Main site of science textbooks
Crowdfunding site for the free
online science textbooks project
Phyiscs of Shaolin Kung Fu
Dr. Jay Maron

Shaolin Abbott:   What is the highest technique you hope to achieve?
Bruce Lee:            To have no technique.


The prime focus of meditation is the topmost vertebra of the spine, the atlas vertebra.

Atlas vertebra

Axis vertebra

Head motion

The atlas and axis vertebra move your head the same way an alt-azimuth mount moves a telescope.

Altazimuth telescope mount
Keck telescope altazimuth mount


Pitch is controlled by the Atlas-Skull joint. (Nodding your head "yes")
Yaw is controlled by the Axis-Atlas joint. (Shaking your head "no")
Roll is controlled collectively by all neck vertebrae.

Suboccipital muscles
Suboccipital muscles

The suboccipital muscles connect the skull, the atlas vertebra, and the axis vertebra.

Head balance

The atlas vertebra is at the center of the head and your eyes and ears are at the same level as the atlas vertebra.

The center of mass of the skull is slightly forward of the contact point between the skull and the atlas vertebra. If the muscles in your neck relax then your head pitches forward. If your back muscles relax then your torso pitches forward. The muscles in the back of your neck and spine act reflexively to prevent you from falling forward. This motion is coordinated with the breathing cycle.

Breathing cycle

Bruce Lee: When the opponent expands I contract, when he contracts I expand, and when there is an opportunity, I do not hit, it hits all by itself.

The thoracic diaphragm is underneath the lungs. When it contracts it creates overpressure in the gut, expands the ribcage, and creates underpressure in the lungs. The lung underpressure brings in air. Air is expelled from the lungs by contracting the rib intercostals.

Pelvic diaphragm

The pelvic diaphragm works in opposition to the thoracic diaphragm. When the thoracic diaphragm contracts the pelvic diaphragm expands and vice versa. This moves the gut cyclically up and down.

Muscle coordination

Breathing is coordinated with skeletal motion to minimize energy expense. For example, when you breathe in your head tends to pitch back and when you breathe out your head tends to pitch forward. It can be done the opposite way but it's less natural.

Every bilaterally symmetric motion is related to the breathing cycle according to the following table:

Inhale                     Exhale
Diaphram contracts         Diaphragm expands
Abdominals expand          Abdominals contract
Gut squashed by diaphram   Gut expands
External intercostals      Internal intercostals
Ribcage expands            Ribcage contracts
Pelvic floor expands       Pelvic floor contracts
Spine muscles contract     Spine muscles release
Head rises                 Head descends
Arms out                   Arms in
Arms rotate thumbs up      Arms rotate thumbs down
Elbows rotate out          Elbows rotate in
Open hand                  Form fist
Feet rotate outward        Feet rotate inward
Knees apart                Knees together
Lower back arches          Lower back sags
Hips rotate forward        Hips rotate back
Daydream                   Focus
Rebalance                  Exertion
High moment of inertia     Low moment of inertia
Discard angular momentum   Discard pressure
Tongue makes "U" shape     Tongue makes flat shape, such as for the letter "L"
Spread fingers into a fan  Bring fingers together like a fin

Rows correspond to opposite directions of a motion, and columns show how they synchronize with the breathing cycle.

Axis cycle

Bilaterally antisymmetric motions are coordinated through the axis vertebra. Motion naturally cycles between the left and right columns in the table below:

Head rotates left                   Head rotates right
Right hand rotates thumbs-up        Right hand rotates thumbs-down
Left hand rotates thumbs-down       Left hand rotates thumbs-up
Right foot rotates right            Right foot rotates left
Left foot rotates right             Left foot rotates left
Right arm rotates out               Right arm rotates in
Left arm rotates in                 Left arm rotates out
Jaw pivots right                    Jaw pivots left
Tongue pivots right                 Tongue pivots left
Eyes pivot right                    Eyes pivot left
Head rolls right                    Head rolls left
Shoulders rotate right              Shoulders rotate left
The axis cycle is structured to most naturally conserve angular momentum. For example, if your right arm rotates thumbs-down then conservation of angular momentum is satisfied if your left arm rotates thumbs-up.
Cycle properties
Cycle      Nexus            Head motion   Motion type                   Goal

Breathe    Atlas vertebra   Pitch         Bilaterally symmetric         Minimize energy expense
Axis       Axis vertebra    Yaw           Bilaterally antisymmetric     Minimize internal angular momentum

Roll-Yaw relationship

When you move your head, it is most natural to coordinate rightward roll with rightward yaw, and leftward roll with leftward yaw. It can be done the opposite way but it's less natural.

The reason roll and yaw are related this way is because the center of mass of your head is forward of the balance point between your spine and skull. If you start from an upright position, roll right, and then stop, then conservation of angular momentum causes your head to yaw right.

Pitch-Yaw relationship

When your head pitches back, it will naturally tend to yaw. For some people it yaws right and for some it yaws left.

If pitching your head back is naturally accompanied by yawing to the right, then pitching your head forward is naturally accompanied by yawing to the left.

If pitching your head back is naturally accompanied by yawing to the left, then pitching your head forward is naturally accompanied by yawing to the right.


The spine consists of a set of curves that function as shock absorbers and as vertical motion for the head.

Cervial vertebrae
Cervical vertebrae

Lumbar vertebrae

Meditation strategy

The prime focus of meditation is developing awareness of the atlas vertebra and how it pivots. The longer you can hold your focus on the atlas vertebra the better. You won't be able to focus forever and so there are other things you can think about, in order of importance:

1)  Pitch       Atlas vertebra and the breathing cycle.
2)  Yaw         Yaw-Pitch relationship.  Axis vertebra and the axis cycle.
3)  Roll        Roll-Yaw relationship.


Syrio: This is not the dance of the Westeros we are learning. The knight's dance, hacking and hammering. This is the Braavos dance, the water dance. It is swift and sudden.

Wire stance

Gordon Liu demonstrates the wire stance in the film "Instructors of Death"
Yip Man

Balance flows from a stance with knees in and heels out, the "wire stance". In this stance all of your leg joints are free to move and you are able to accelerate in any direction. Imagine that you are a goalie and you are braced to defend a shot.

Combat stance

Bruce Lee: "Balance is the all-important factor in a fighter's attitude or stance. Without balance at all times, he can never be effective."

Bruce Lee: "In general for athletic contests, a preparatory stance a coiled, or semicrouched posture and a lowered, forward center of gravity. with the bending of the forward knee, the center of gravity moves forward a little. For general readiness, the lead heal usually remains just touching even after the knees bend. Slight ground contact of the heel aids in balance and decreases tension."

Bruce Lee: "The center of gravity kept under delicate and rapid motion are characteristic habbits of athletes in games that require sudden and frequent changes of direction."

Video:     Gordon Liu demnstrates the wire style at time=4:37.     Bass guitarist Robert Trujillo.     Yip Man.     Violinist Mikhail Kopelman

Jascha Heifetz

Jascha Heifetz masterclass.     In this clip at time=5:45-6:15 Heifetz emphasizes the importance of balance and posture. You should be able to descend in altitude without your head pitching forward, and your weight should be on your heels.

Your posture should be such that you are able to harness your spine curvature as a shock absorber.

Low center of mass

You should be able to balance without your limbs rising unnecessarily.

Ray Park demonstrates how to maneuver with a low center of mass and with his arms down.

Gil Shaham plays with a low bow arm.

Byamba vs. Kelly.     Byamba wins this sumo match because he keeps his arms and center of mass lower than his opponent.

In a sumo match, the more you lean forward the more forward force you can project, but if you lean too far forward you get faceplanted.

Faceplant defense

Barry Sanders, running back.


Bruce Lee: "Practice your balance by standing on one foot."

Bruce Lee: "Always leave the space of a natural step between your feet. By doing so, you are braced and never standing just on one foot."

You should be able to balance effortlessly on either foot, and when you move you should be able to get as much done as possible with each step.

The heart and the viola are on the left side of your body. Your center of balance should be planted on your right foot with your left foot free to float so that the left side of your body is free to maneuver the instrument. Avoid leaning unnecessarily to the left.

If your feet lose their freedom to move then you start to sway. Avoid swaying. Also avoid moving the viola unnecessarily. Gain an awareness of the motion of the scroll and keep it under control.

Leonidas Kavakos plays without sway and the instrument is stable. His head is also centered. If your head turns to the left then you can't integrate the motion of your head with your body.

Michael Shulman

Leonid Kogan

Yann Tiersen, time=3:40

When the body and the viola are stable then the right arm and bow are free to move independently. Otherwise the motion of the body, the viola, and the bow become entangled.

Erika Gray and right-foot balance


Axial skeleton
Appendicular skeleton

Long bones
Flat bones
Short bones
Irregular bones

Wave impedance

Slow motion baseball pitch

In a baseball pitch, the motion starts from the feet and then progresses to the hips, the torso, the shoulder, the upper arm, the lower arm, the wrist, and then to the fingers. This maximizes the speed that can be delivered by the fingers.

A whip is tapered so that it gets progressively thinner as it goes from the handle to the tip, just like your arm. As a pulse travels down the whip the decrease in thickness causes the amplitude to increase. If the taper is smooth then the reflection of energy in the pulse is minimized and the maximum energy is delivered to the tip. If there is a sharp change in thickness then part of the energy in the pulse is reflected. This is a consequence of the phenomenon of "wave impedance".

In standard viola technique the index finger releases from the bow at the apex of the downbow. If you do this then you can't harness the whip. To harness the whip the first knuckle has to be wrapped firmly around the bow at the apex of the downbow so that the bow can be pulled with force. Also, the wrist has to be straight or bent outward rather than collapsing inward.

The Snake Fist

The fingers and thumb should work together without exerting torque on the forearm.

When holding the bow, the index finger and thumb should be able to make contact with a minimum of stress.

You should be able to focus your awareness on tip of your index finger, analogous to the tip of the sword. You should also be able to focus your awareness on the tip of the bow.

Snake foot: Jackie Chan vs. Ken Lo.     Watch Ken Lo's feet.

The Drunken Levitating Viola

Andrew Luck fumbles the football

Syrio:  It is not a great sword that is needing two hands to swing it.
        The grip must be delicate.
Arya:   What if I drop it?
Syrio:  The steel must be part of your arm!
        Can you drop part of your arm?  No!

The sword

Li Mu Bai: Real skill comes without effort. No passion without restraint. You must learn to hold it in stillness.

Gain an awareness of the motion of the tip of the sword and be able to maneuver it with minimalist effort. The sword should move without interference from the body. This is especially critical for a viola bow because it is so light.

Li Mu Bai training with a sword
Li Mu Bai vs. Jade Fox and Jen
Li Mu Bai vs. Jen
Jet Li and the drunken sword

Instrument balance

The instrument should be incorporated into your balance.

*    At time=1:05 Bruce Lee integrates the balance of his arms into his body.
*    Marie Daniels illustrates capturing the balance of the viola.
*    Hilary Hahn
*    Sammo Hung vs. Wu Jing. Techniques for gaining awareness of the motion of a sword or staff.
*    Marie Daniels: techniques for gaining awareness of the motion of a bow.

Drunken Master

Bruce Lee: "One should seek good balance in motion and not in stillness."

Bruce Lee: "Balance is the control of one's center of gravity plus the control and utilization of body slants and unstable equilibrium, hence gravity pull to facilitate movement. So, balance might mean being able to throw one's center of gravity beyond the base of support, chase it, and never let it get away."

Bruce Lee: "The short step and the glide, as contrasted with the hop or cross step, are devices to keep the center of gravity. When it is necessary to move rapidly, a good man takes small enough steps so that his center of gravity is rarely out of control."

Jackie Chan from 'Drunken Master'

The Eagle Claw

Leopold Auer: Place the four fingers on the four strings -- the first on the F of the E-string; the second on the C of the A-string; the third on the G of the D-string, and the fourth on the D of the G-string. Do not raise any one of the fingers until all four are resting on the notes mentioned.When the fingers are all in place, exercise each finger separately, raising it and allowing it to drop back into place several times in succession. Begin with the second finger, then take the fourth, and then the first, and, finally, the third. The exercise should be carried out in such wise that the fingers not engaged remain on the strings. This exercise, persisted in, is certain to accomplish two valuable things -- absolute correctness of finger position, especially as regards the thumb, and increased finger strength.

The violinist and teacher Leopold Auer, in his book "Violin Playing as I Teach It" (1920), advised violinists to practise playing completely without vibrato, and to stop playing for a few minutes as soon as they noticed themselves playing with vibrato in order for them to gain complete control over their technique.

The fingers should coil without exerting torque on the forearm.

The joints of the fingers should curl naturally so that all joints are engaged. This gives you more layers of stability. When a finger is not on the string it shouldn't drift too far from the string.

If you play with the tip of the finger contacting the string rather than the flat part then you can harness all of the joints in your fingers for stability.

The fingers should approach the string from right to left, with the tip of the finger making contact with the string. This minimizes unnecessary disruption of the vibration of the string when the fingers make contact.

You should be able to exert a strong downward force on the string with your fingertip without any part of the rest of your left hand touching the viola. You should also be able to shift positions in this manner.

David Garret, left hand technique

The Metal Head

The less your head moves the more you can think. It's also important for the
head to move with the body without pivoting the atlas or axis vertebrae.
Leonidas Kavakos
Karianne Brouwer
Gil Shaham
Sarah Tuke

Metal is best sung with zero vibrato (Osbourne, Metallica).


Mathematical focus requires being able to keep the head centered and down. Focus is lost when the head pops up. A sharp exhale can be used to maintain focus and a sharp inhale is a loss of focus.

Avoid breathing with your mouth if possible.

Wizard battle from "Conan the Barbarian"


Maneuverability requires maintaining a low center of mass, with the knees deeply bent.

Barry Sanders the running back

From Wikipedia: Though there were concerns about his size, it turned out these concerns were unfounded. Sanders was far too quick for defenders to hit solidly on a consistent basis, and too strong to bring down with arm tackles. Though short at 5 foot 8, his playing weight was 203 lb (91 kg). Sanders had unusual explosiveness.


There is an analogy between elasticity and sports. An athlete should have both agility and endurance and there tends to be a tradeoff between the two. For materials, the tradeoff is between tensile strength and tensile modulus.

Endurance    =  Energy  / Mass  =  Strength / Density
Agility      =  Power   / Mass
Stiffness    =                  =  Modulus  / Density

Wrestleness  =  Agility   * Endurance  =  Power   * Energy   / Mass2
Swordness    =  Stiffness * Endurance  =  Modulus * Strength / Density2
"Wrestleness" reflects a synthesis of endurance and agility, and "swordness" reflects a synthesis of tensile modulus and tensile strength. Both give emphasis to being lightweight.
Mass         =  M
Volume       =  Υ
Density      =  D  =  M / Υ
Strength     =  Ρ                          Tensile strength. Pressure to break the material
Modulus      =  ΡM                         Young's modulus
Endurance    =  Ε  =  Ρ  / D  =  E / M
Stiffness    =  ΕM =  ΡM / D

Modulus      =  ΡM                         Young's modulus
Endurance    =  Ε  =  Ρ  / D  =  E / M
Stiffness    =  ΕM =  ΡM / D
Energy       =  E  =  Ε M

Time         =  T                           Wave time for the lowest overtone
Power        =  P  =  E / T
Agility      =  Ξ  =  P / M
Wrestleness  =  Ω  =  Ξ  Ε    =  P  E / M2
Swordness    =  Ψ  =  ΕM Ε    =  ΡM Ρ / D2
Time is analogous to malleability.
Frequency    =  F
Time         =  T  =  E / P  =  F-1
Malleability =  ε  =  Ρ / ΡM                Fractional elongation at the breaking point
Rigidity     =  ξ  =  ε-1
Wrestleness  =  Ω  =  Ξ  Ε  =  Ξ2 T  =  Ε2 F
Swordness    =  Ψ  =  ΕM Ε  =  ΕM2 ε  =  Ε2 ξ

Maximum stress

For a stringed instrument, the most difficult stretch for the left hand is to play with the pinky on the lowest string.

In this clip, before Janine Jansen begins the concerto, she places the pinky of the left hand on a high note on the g-string, the point of maximum stress, and then lets the stress dissipate.

Bruce Lee: "Jeet Kune Do avoids the superficial, penetrates the complex, goes to the heart of the problem and pinpoints the key factors."

Identify the most difficult element of a problem and make that the central focus.

Bruce Lee: "Let yourself go with the disease. Be with it, keep company with it - this is the way to be rid of it."


"The Beast" from "Kung Fu Hustle": "In the world of kung fu, speed determines the winner."

Bruce Lee: "It's not daily increase but daily decrease -- hack away at the unessentials!"

Bruce Lee: "Well executed movement meas the nervous system has been trained to the point where it sends impulses to certain muscles, causing these muscles to contract at exactly the proper fraction of a second. At the same time, impulses to the antagonistic muscles are shut off, allowing those muscles to relax. Properly coordinated impulses surge with just the exact intensity required and they stop at the exact fraction of a second when they are no longer needed."

When the biceps muscle contracts, the speed of the motion is determined both by how fast the biceps contracts and by how fast the triceps expands. You have to generate a sharp nerve impulse that contracts the biceps and you also have to inhibit contraction impulses to the triceps.

The triggered release of muscle pressure or tension can generate a sharper impulse than muscle contraction. For example, the thorasic diaphragm delivers tension energy to the intercostal muscles at a slow rate and then the energy can be released as a sharp pulse by releasing the intercostals. The pulse is then channeled to whatever part of the body needs to move.

While moving, avoid unnecessary changes in direction of any body part because this generates muscle noise.

Muscle mechanics

Muscles have their highest speed when they are under tension. You can use a backswing to preload your muscles with tension.

Muscle power is proportional to muscle force times muscle speed. If a muscles is exerting maximal force at zero speed or zero force at maximal speed then no power is being generated. The speed that maximizes power is 1/3 of the maximal speed.

A gear system allows you to change the speed at which force is applied so that you can use a natural speed.

The Vision:   It's terribly well balanced.
Thor:         Well if there's too much weight you lose power on the swing.

Principle of least action

Brachistochrone curve

The brachistochrone curve delivers each ball to the endpoint the fastest. Also, the time to reach the endpoint is independent of the starting point.

You are always using more muscle motion to accomplish a task than you need to, and you are always moving your fingers and the bow more than you need to. Use a strategy that involves the least possible motion and the fewest possible changes in direction.

To find the path of least motion you will have to practice moving in slow motion. To play fast you have to practice slow.

Video: Metronome scene from the film "The Red Violin"

Chuck Norris follows the "Principle of Most Action". The SI unit of action is the "Chuck".

The One Inch Downbow

Pai Mei:   Can you do that?
Beatrix:   I can, but not that close.
Pai Mei:   Then you cannot do it!
Practice initiating a motion with maximum suddenness and with minimal backswing.

If you attempt a suddent jump then the motion tends to overshoot, which is an aspect of the Gibbs phenomenon. You can use this overshoot to preload your muscles with tension.

The 5 Point Palm Exploding C String Technique

The deadliest bow in all of martial arts.

Become connected to the resonances of the instrument. Practice listening to the sound of the body of the instrument. Play without vibrato and channel energy into the resonances.

Phase lock

The elements of the breathing cycle or axis cycle can be synchronized so that the motions are in phase.

Sumo. (starting at time=5:40)


In standard viola technique the wrist collapses inward at the apex of the downbow, If you do this then you can't connect the rotation of your hand with your arm and body, and you can't deliver power through your fingertips. To do this you have to play with a straight wrist.

A collapsed wrist creates an impedance mismatch between your arm and your hand, preventing energy from flowing between them.

Gordon Nikolitch plays with a straight wrist.


When playing a piece of music you should be able to start and stop at will without it disrupting your internal timing.

Perelman, time = 5:00 - 5:20

Bruce Lee: "An occasional change of pace should be included that employs different movements, and to some degree, different muscle fibers."


Bruce Lee: "Experiments indicate that auditory cues, when occuring close to the athlete, are responded to more quickly than visual ones. Make use of auditory clues together with visual clues, if possible. Remember, however, the focus of attention on general movement produces faster action than focus on hearing or seeing the cue."

36th Chamber of Shaolin, time=1:04.    Moving your eyes without moving your head.

Practice moving your eyes between the conductor, the concertmaster, your section principal, and the score, so that you can take in as much information as possible with a single glance.

Bruce Lee: "For most rapid perception, attention must be at its maximum focus on the area of the thing to be perceived."

If your eyes are open your brain has to work to hard to process the information and it is difficult to meditate. Auditory stimulus is better for meditation.

Animal styles

Shaolin monks imitating frogs
Arya chasing a cat

Arya     Sirio says a water dancer can stand on one toe for hours.
Eddard   It's a hard fall down these steps.
Arya     Sirio says every hurt is a lesson and every lesson makes you better.
         Tomorrow I'm going to be chasing cats.
Eddard   Cats?  Sirio says?
Arya     He says every swordsman should study cats.
         They're as quiet as shadows and as light as feathers.
         You have to be quick to catch them.


Your shoulder blades should be free to move independently.

Maintain focus while moving.

David Blaine

Stood for 35 hours on a pillar
Electrified for 72 hours
Entombed in a coffin for 7 days

Underwater for 17 minutes and 4 seconds
Encased in ice for 63.7 hours
Encased in a hanging box for 44 days without food

One has to develop endurance for meditation.

Damage control

A-10 Warthog
Cannon firing

Still flew

The mission of an A-10 Warthog is to fly close to the ground and shoot at tanks. It's designed to take heavy damage and still function. It can fly with one engine, one tail, one elevator, and half of one wing missing.

Crew           1
Empty weight   11.3 tons
Max weight     23 tons
Engines        2 engines, 40.3 kNewtons each
Max speed      200 meters/second
Stall speed    63 meters/second
Ceiling        13.7 km
Cannon         30 mm Gatling gun with 7 barrels, 1174 rounds, and 65 rounds/second
The A-10 is exceptionally tough, being able to survive direct hits from armor-piercing and high-explosive projectiles up to 23 mm. It has double-redundant hydraulic flight systems, and a mechanical system as a back up if hydraulics are lost. Flight without hydraulic power uses the manual reversion control system; pitch and yaw control engages automatically, roll control is pilot-selected. In manual reversion mode, the A-10 is sufficiently controllable under favorable conditions to return to base, though control forces are greater than normal. The aircraft is designed to fly with one engine, one tail, one elevator, and half of one wing missing.

The cockpit and parts of the flight-control system are protected by 1,200 lb (540 kg) of titanium aircraft armor, referred to as a "bathtub". The armor has been tested to withstand strikes from 23 mm cannon fire and some strikes from 57 mm rounds. It is made up of titanium plates with thicknesses from 0.5 to 1.5 inches (13 to 38 mm) determined by a study of likely trajectories and deflection angles. The armor makes up almost 6 percent of the aircraft's empty weight. Any interior surface of the tub directly exposed to the pilot is covered by a multi-layer nylon spall shield to protect against shell fragmentation. The front windscreen and canopy are resistant to small arms fire.

The A-10's durability was shown on 7 April 2003 when Captain Kim Campbell, while flying over Baghdad during the 2003 invasion of Iraq, suffered extensive flak damage. Iraqi fire damaged an engine and crippled the hydraulic system, requiring the aircraft's stabilizer and flight controls to be operated via the 'manual reversion mode'. Despite this damage, Campbell flew the aircraft for nearly an hour and landed safely.

To reduce the likelihood of damage to the A-10's fuel system, all four fuel tanks are located near the aircraft's center and are separated from the fuselage; projectiles would need to penetrate the aircraft's skin before reaching a tank's outer skin. Compromised fuel transfer lines self-seal; if damage exceeds a tank's self-sealing capabilities, check valves prevent fuel flowing into a compromised tank. Most fuel system components are inside the tanks so that fuel will not be lost due to component failure. The refueling system is also purged after use. Reticulated polyurethane foam lines both the inner and outer sides of the fuel tanks, retaining debris and restricting fuel spillage in the event of damage. The engines are shielded from the rest of the airframe by firewalls and fire extinguishing equipment. In the event of all four main tanks being lost, two self-sealing sump tanks contain fuel for 230 miles (370 km) of flight.

Practice under nonideal circumstances.

When the going gets tough you will need an arsenal of damage control moves. An hour-long symphony is a long time to be sitting in a chair and you'll need methods to dissipate tension.

Jet Li vs. Billy Chow.    Exhibit of damage control moves


The height of Manhattan skyscrapers is determined by the quality of the bedrock.

Etudes and scales are designed to develop fundamentals.


Wire stance
Breathing cycle, walk cycle, axis cycle
Spine curvature
Ulna point and tibia point

The dark side

Avoid cheated technique and focus on fundamentals. Avoid hasty gains.

Your defenses must be as flexible and inventive as the Arts you seek to undo.
It's my job to think as dark wizards do. When it comes to the dark arts I believe in a practical approach. You need to know what you're up against.
You can exist without your soul but there's no chance of recovery. You'll just exist. An empty shell.
The enemy has only images and illusions behind which he hides his true motives. Destroy the image and you will break the enemy.

Vibrato, an invention of the Sith

In the baroque age violists played with a pure, vibratoless tune, using bow speed rather than vibrato for expressivity. After the baroque age, an epidemic of vibrato emerged and is still with us, especially at Juilliard and Lincoln Center. Vibrato obstructs the resonances of just intonation.

"There are performers who tremble consistently on each note as if they had the permanent fever" - Leopold Mozart, 1756

The violinist and teacher Leopold Auer, in his book "Violin Playing as I Teach It" (1920), advised violinists to practise playing completely without vibrato, and to stop playing for a few minutes as soon as they noticed themselves playing with vibrato in order for them to gain complete control over their technique.


What do you call a fast man that can't catch?               A cornerback
What do you call a fast man that can't tackle?              A wide receiver
What do you call a fast man that can't tackle or catch?     A sprinter
What do you call a fast man that can tackle and catch?      A rugby player
What do you call a slow man man that can't tackle
  or catch, but is badass in intangible ways?               A rugby player


Raymond Berry was a wide receiver who played for the Colts with Johnny Unitas. He would train by going to a football field by himself and go through the motions of an entire game, play by play, stopping to think from time to time. He and quarterback John Unitas also regularly worked after practice and developed the timing and knowledge of each other's abilities that made each more effective. They developed a telepathic connection so that they were able to freely improvise during games.

Berry played in 6 pro bowls and was selected to the Pro Football Hall of Fame. During his career he dropped only 2 passes and fumbled only twice. He achieved this success in spite of the fact that he was not a fast runner, usually a necessity for a wide receiver, relying instead on mental focus and rigorous training.

Terry Bradshaw played quarterback for the Pittsburgh Steelers and won 4 super bowls. He called his own plays throughout his football career, in contrast with most teams where the plays are called by coaches.

Kirk       Do you hear that?
Odona   Yes. Sounds like an engine.
Kirk       Ship's engines don't make that kind of sound.
Odona   But there is something wrong with the equipment. Could that be it?
Kirk       I know every sound that this ship might make. It's coming from outside.

Hunt for Red October

Crewman #1: Torpedo! The Americans are shooting at us again! Crewman #2: Pitch is too high. The torpedo's Russian.


Bruce Lee emphasizes avoiding repetitive training because then you are substituting high-precision circuitry for low-resolution circuitry. Change the timing from time to time.


Bruce Lee: "In Jeet Kune Do, it's not how much you have learned, but how much you have absorbed from what you have learned. It is not how much fixed knowledge you can accumulate, but what you can apply livingly that counts. 'Being' is more valued than 'doing'."

Bruce Lee: "Forms are vain repetitions which offer an orderly and beautiful escape from self-knowledge with an alive opponent."

Bruce Lee: "Set patterns, incapable of adaptability, of pliability, only offer a better cage. Truth is outside of all patterns."

Matrix: Morpheus vs. Neo
Neo:    I know kung fu!
Morpheus:   Show me.


Bruce Lee: "A good martial artist does not become tense, but ready. Not thinking, yet not dreaming. Ready for whatever may come. When the opponent expands, I contract; and when he contracts, I expand. And when there is an opportunity, 'I' do not hit, 'it' hits all by itself."

Bruce Lee: "The great mistake is to anticipate the outcome of the engagement; you ought not to be thinking of whether it ends in victory or defeat. Let nature take its course, and your tools will strike at the right moment."

Bruce Lee: "Train yourself to cut down unnecessary choice reactions (minimizing yourself naturally) while giving your opponent a variety of possible responses."

Bruce Lee: "A good man is continually tring to force his opponent into the slower, choice-reaction situation."

Bruce Lee: "The man who is clear and simple does not choose. What is, is. Action based on an idea is obviously the action of choice and such action is not liberating. On the contrary, it creates further resistance, further conflict. Assume pliable awareness."

Bruce Lee: "Progression from volition to reflex control is when an athlete's awareness is shifted from small details (mechanical performance) to larger ones, and finally to the whole action, without any thought given to any single part."

Bruce Lee: "Speed of perception is somewhat affected by the distribution of the observer's attention - fewer separate choices, faster action. When the cue to be recognized is one of several, each of which requires a different response, the time is lenghened. Choice reaction is longer than simple reaction. This is the basis for training the tools in terms of neurophysiological adjustment toward instinctive economy. Instinctive movement, being the simplest, is the quickest and most accurate."

Bruce Lee: "The habbit of diffusing the attention to a wider area helps the offensive passer see openings more quickly."

Bruce Lee: "Don't think, is like a finger pointing a way to the moon. Don't concentrate on the finger or you will miss all that heavenly glory!"

Bruce Lee: "Do not deny the classical approach, simply as a reaction, or you will have created another pattern and trapped yourself there."

Bruce Lee: "While being trained, the student is to be active and dynamic in every way. But in actual combat, his mind must be calm and not at all disturbed. He must feel as though nothing critical is happening. When he advances, his steps should be light and secure, his eyes not fixed and glaring insanely at the enemy. His behavior should not be in any way different from his everyday behavior no change taking place in his expression, nothing betraying the fact that he is engaged in mortal combat."

Yip Man: "You must know the source of the water from which you drink."

Poker face

Bruce Lee: "A correct posture does three things: It insures for the body and its several members a position which is most mechanically favorable for the next move. It enables one to maintain a 'poker body', a body that reveals no more of its intended movements than a 'poker face' reveals the cards of a player. It puts the body under that particular tension or degree of tonus which will be most favorable to quick reaction and high cooordination."

Cloak your motion so that it is unpredictable and practice predicting your opponent. Practice predicting your opponent's hands from watching his feet.

Sammo Hung, time=1:55

James Bond vs. Le Chiffre from 'Casino Royale'

The singing sword

Von Karman vortices
Karman vortex street caused by wind flowing around the Juan Fernandez Islands off the Chilean coast.
Canary islands

Air moving past a cylinder generates "Von Karman vortices", which have a frequency given by

V  =  Velocity of the cylinder
D  =  Diameter of the cylinder
S  =  Dimensionless Strouhal number
   =  1/5  for most situations
F  =  Frequency of vortices
   =  S V / D
Typical values for a moving bow are
V  =  1 meter/second
D  =  .005 meters (typical value for the cross-section of the bow hair)
F  =  40 Hertz
If you hold a bow to the wind or wield it like a sword you can feel the Von Karman vortices. The bowhair is lightweight and easily activated by the vortices. This allows you to perceive the bow's speed as a sound frequency, which is more precise.

The frequency of vortices from wind moving past your body at 1 m/s is

F  =  (1/5) * (1 m/s) / ((1/3) meters)
   =  3/5 Hertz

Standard violin technique

Mimas, a moon of Saturn
Not a moon

Collapse the wrist at the downbow

The index finger leaves the bow at the downbow

Relentless vibrato

The bow is perpendicular to the string and moves in 1 dimension

Arpeggios are played near the tip

The pinky is placed flat on the string

The wrist pivots like Mr. Miyagi's "paint the fence" technique

Motion of the bow along the string is generated by the elbow and string crossings are generated from the shoulder.

Worship at the shrine of Suzuki and Galamian.


Use shoes with thin soles so your feet can feel the ground. You also want your feet as close as possible to the ground.

Yehudi Menuhin

Yehudi Menuhin's violin lecture

This lecture is like a review of the physics discussed in this text. Menuhin's presention is richly informed with the physics of modes, resonances, and the breathing cycle.

Elements from the lecture:

Isolate each motion into its 3 component dimensions and then reassemble it into 3 dimensions.

Balance. Be long so that your joints are engaged. Stance with feet at shoulder width. Be fluid not static.

Practice standing on one foot.

Practice putting your hands on the ground and rising back up. Shake on the way up.

The head is a limb.

Motion starts from the body and progresses in sequence to the arm, hand, and bow.

Isolate the motion of the shoulders.

Flap your arms and link them to the breathing cycle.

Rotate your arms at the shoulders in all possible combinations.

Isolate skeletal overtones. Be aware of which body parts are in contact with external objects and which are free.

Be able to pull the shoulders down without them bouncing back up.

Mental focus is as important as a fluid body.

Don't hold the stick. Just keep the stick from falling.

Avoid collapsed joints. (Loss of midrange)

When holding the bow, the fingers should be able to exert forces opposite each other.

One does not apply pressure to the bow. One instead allows the arm to fall onto the bow, resisted by the fingers. When one plays a downbow one should have control over the descent of the right hand. Play with the weight of the hand.

Practice going back and forth between separating your fingers into a fan and combining your fingers into a fin. Try this with your knuckles as well.

When the right hand grasps the bow there should be space between the fingers so that each finger can act independently without contact with adjacent fingers.

When the right hand grasps the bow the fingers are angled so that the motion of the bow coincides with the degrees of freedom of the knuckles. This implies rotating the right hand in the thumbs-down direction. The "Russian bow grip" uses an extreme angle.

Practice the side-to-side wrist motion.

Practice waving the bow when it is both close to your body and far from your body.

Move your right hand like a whip, where the impulse is generated from your body.

The violin rests on the collarbone so that the shoulder is free to move.

Avoid supporting the violin with the left hand.

From time to time, stop and test to see if all joints are free to move.

Develop the curl of the left fingers.

The left fingers should press down on the string vertically.

When a left finger is planted the hand and arm should be free to move.

When the bow is planted on the string the right hand and arm should be free to move.

Add backswing and follow through to your motion. Make elliptical and figure-8 motions.

Count multiple beats for each motion (2, 4, 3, 8, etc). This adds resolution to your timing.

Vocal muscles

The soft palate separates the nasal cavity from the mouth. When you yawn the soft palate rises, closing off the nasal cavity from the throat. When you swallow the soft palate falls, closing off the mouth from the throat.

The pharynx muscles constrict the throat. When they are relaxed the throat expands. Loosening the tongue also expands the space in the throat. To create space in your throat imagine you're swalling a golf ball.

Try keeping your tongue in contact with the lower part of your mouth without it rising to the top of your mouth.

The diagastric muscle moves the jaw backwards. It starts from the chin, loops through the hyoid bone, and connects to the skull.

There is no consonant or vowel that requires the teeth to be together.

Lowering the larynx expands the vocal cavity and makes it more resonant.

The larynx lowers when you inhale and rises as you exhale.

The larynx lowers when you say the letter "L" and rises when you yawn.

The word "cheese" activates the buccinator muscles and the word "choose" relaxes them.

If you go back and forth between "cheese" and "choose" your mouth moves like a quadrupole, which is characteristic of sphincter muscles.

The aryepiglottic sphincter (henceforth "larynx sphincter") is just above the larynx and is below the pharynx. It is important for singing loudly. You can improve the power and resonant quality of your throat by either widening the pharynx or constricting the larynx sphincter. It is activated if you yawn or cackle like a villain.

The tongue connects to the lower chin and most of the mass of the tungue is below the teeth.

The letters "L", "N", and "K" are difficult to pronounce without lowering the jaw. Try saying the words "Nickle", "Lincoln", and "Klink" while keeping your teeth together.

Try putting your fingers on your cheeks and feeling the motion of your jaw while you say the letters "A", "B", "C", etc. Try thinking about saying the letters without saying them.

Dipthongs are a concatenation of two different vowels.

An aspirated consonant is one that involves expelling air without creating a pitch.

Aural illusions

Meara O'reilly's set of illusions
Diana Deutsch Speech-to-song illusion
Sinewave speech synthesis
Shepard tones


An uppercut requires a low elbow. Power is delivered from the ground through the right foot into the right hand.

Mark Hunt


Beth Blackerby's youtube channel on violin technique
Amy Walker's youtube channel on vocal technique
Yehudi Menuhin's lecture on violin technique
Jascha Heifetz's lecture on violin technique
Bellydancing   (treat the body as a set of overtones).


Bruce Lee vs. Chuck Norris
Jet Li vs. Billy Chow
Jet Li vs. Chin Siu-ho
Sammo Hung vs. Wu Jing
Jackie Chan vs. Ken Lo
Jackie Chan Drunken Master training
Jackie Chan vs. Jet Li
Gordon Liu in 'Instructors of Death'
Crouching Tiger Hidden Dragon, Bamboo forest scene
Chow Yun Fat with a sword
Gordon Liu in 'The 36th Chamber of Shaolin'
Glen Levy Ben Askren Pai Mei and the 5 point palm exploding heart technique Sammo Hung in "Enter the Fat Dragon"

Styles for the mind-body system

History of kung fu

 -776       Hercules establishes the Olympic games, with wrestling as a sport
 -648       The sport of "Pankration" is introduced to the Olympic Games. Similar to MMA.
 -536 -520  Milo of Croton dominates Olympic wrestling
 -450       Gautama Buddha develops the art of meditation
  464       Batuo, a monk from India, founds the Shaolin Temple
  500       Bodhidharma, a Buddhist monk, teaches at the Shaolin temple
 1600       Sumo emerges in Japan
~1700       Shaolin temple destroyed by the Chinese Emperor.
            The monks who escaped spread Shaolin kung fu throughout China.
            These monks were:
            Ji Sin  - Developed Tiger Crane style
            Ng Mui  - Developed Wing Chun
            Bak Mei - Known as "Pai Mei" in kung fu films. Appears in "Kill Bill"
~1700       Fong Sai Yuk.  Portrayed by Jet Li in the "Fong Sai Yuk" film series.
 1847 1924  Wong Fei Hung.  Master of Hung Gar style. Portrayed by Jet Li in
            the "Once Upon a Time in China" film series.
 1860 1938  Jigoro Kano.  Developed Judo and taught it to Mitsuyo Maeda and
            Moshe Feldenkrais
 1868 1910  Huo Yuanjia.  Portrayed by Jet Li in the film "Fearless"
 1869 1955  F.M. Alexander.  Developed "Alexander Technique"
 1893 1972  Yip Man.  Practitioner of Wing Chun.  Teacher of Bruce Lee
 1904 1984  Moshe Feldenkrais.  Physicist and wrestler.
            Developed the "Feldenkrais Method".
 1912 1979  Simon Yuen
 1917       Mitsuyo Maeda teaches Judo to the Gracie family.
            Helio Gracie subsequently develops Brazilian Jiu-Jitsu
 1922       Usui develops Reiki
 1933       Jigoro Kano trains Feldenkrais in Judo
 1940 1973  Bruce Lee
 1940       Chuck Norris
 1949       Feldenkrais trains with Alexander
 1951       Masahiko Kimura vs. Helio Gracie
 1952       Sammo Hung
 1954       Jackie Chan
 1955       Gordon Liu
 1963       Jet Li
 1963       Michelle Yeoh
 1963       Donnie Yen
 1970       Shaw Brothers Studios begins mass-producing kung fu films, featuring the "Venom Mob"
               of actors, including Lo Mang, Kuo Chui, Chiang Sheng, and Dick Wei
 1975       Bruce Lee's "Tao of Jeet Kune Do" published
 1975       Yehudi Menuhin records a BBC show where he teaches kung fu fundamentals
            for the violin
 1975-      The University of Iowa wins 23 national team wrestling championships
 1993       Age of Mixed Martial Arts begins when Royce Gracie wins a tournament
            consisting of fighers with diverse styles.
            Legions of collegiate wrestlers transition to MMA and dominate
 2000       Kazushi Sakuraba vs. Royce Gracie
 2008       Michael Shulman busks New York City with kung fu violin
 2009-      Ben Askren dominates MMA with wrestling
 2010       Brock Lesnar vs. Cain Velasquez
 2012       Ronda Rousey vs. Miesha Tate
 2013       Ben Askren signs with One-FC.  The UFC declines in significance
 2015       Daniel Cormier vs. Jon Jones

Interviewer:  Other than wrestling what would you say the best base to be a
              successful MMA fighter would be?
Ben Askren:   Wrestling


Mongolian wrestling
College-style wrestling

All wrestling styles share the same fundamentals. Skill in one style translates immediately to skill in other styles.

The Midwest is the center of the wrestling universe. Rural Midwesterners have good balance, especially the farmers.

           National Championships

Oklahoma State   34
Iowa             23
Iowa State        8
Oklahoma          7
Penn State        5
Minnesota         3
Ohio State        1
Arizona State     1
Michigan State    1
Cornell College   1
Indiana           1

Mixed martial arts

Among past and present UFC champions, almost all are wrestlers.


Cain Velasquez      Wrestled at Arizona State and placed 4th nationally.
Junior Dos Santos   Brazilian jiu-jitsu
Shane Carwin        Wrestled for Western State College and won the
                    NCAA Division II national championship
Frank Mir           Wrestled for Bonanza High School in Nevada and won the
                    state championship
Brock Lesnar        Wrestled for Minnesota and won the national championship
Antonio Nogueira    Judo and Brazilian jiu-jitsu
Randy Couture       Wrestled for Oklahoma State and was twice runner up at the
                    national championships
Tim Silva           Wrestled for Ellsworth High School
Andrei Arlovski     Silver medal at the sambo world championships
Ricco Rodriguez     Brazilian jiu-jitsu world champion
Josh Barnett        Wrestled for Ballard High School
Kevin Randleman     Wrestled for Ohio State and won the national championship twice
Bas Rutten          Muay Thai kickboxing
Maurice Smith       World kickboxing champion
Mark Coleman        Wrestled for Ohio State and won the national championship
Light heavyweight
Daniel Cormier      Wrestled for Oklahoma State and placed 2nd at nationals.
                    5-time national freestyle wrestling champion.
                    4th place at the 2004 Olympics.
Jon Jones           Wrestled for Iowa Central Community College and won the
                    Junior College national championship
Mauricio Rua        Wrestling and Brazilian jiu-jitsu
Lyoto Machida       2-time Brazilian sumo champion
Rashad Evans        Wrestled for Michigan State and placed 3rd at the Big Ten
Forrest Griffin
Quinton Jackson     Wrestled for Raleigh-Egypt High School in Tennessee and placed
                    5th at the state championships
Chuck Liddell       Wrestled for California Polytechnic State University
Randy Couture       Wrestled for Oklahoma State and was twice runner up at the
                    national championships
Vitor Belfort       Brazilian jiu-jitsu

Tito Ortiz          Wrestled for Huntington Beach High School and placed 4th
                    at the California state championships
Frank Shamrock      Submission fighting
Chris Weidman       Wrestled for Hoffstra and placed 3rd at the national championships
Anderson Silva      Judo, Brazilian jiu-jitsu
Rich Franklin       Brazilian jiu-jitsu
Evan Tanner         Wrestled for Caprock High School and won the Texas state
                    championship twice
Murilo Bustamante   Judo, Brazilian jiu-jitsu world champion
Dave Menne          Wrestled for Forest Lake Area High School, Minnesota
Robbie Lawler       Wrestled for Bettendorf High School, Iowa, where he received
                    All-State honors
Johnny Hendricks    Wrestled for Oklahoma State and won the national championship
Carlos Condit       Wrestled for Cibola High School, New Mexico
Georges St-Pierre   Wrestling, Hockey, Brazilian jiu-jitsu
Matt Serra          Wrestling, Brazilian jiu-jitsu
Matt Hughes         Wrestled for Eastern Illinois University and placed 5th at
                    the national championships
B.J. Penn           Brazilian jiu-jitsu
Carlos Newton       Wrestling, judo, Brazilian jiu-jitsu
Pat Miletich        Wrestled for Bettendorf High School, Iowa
Ronda Rousey        Silver medal at the 2007 Judo world championships
Joanna Jedrzejczyk  Won 6 Muay Thai world championships
Carla Esparza       Wrestled for Menlo College and was twice an All-American
Ben Askren wrestled for the University of Missouri where he was twice national champion. He is also a former Bellator champion and is currently the One-FC champion. Askren is undefeated in MMA, using wrestling to dominate his opponents.

Wrestling shoes

Wrestling shoes have thin soles for a firm connection to the ground. Avoid thick basketball shoes. Lateral maneuverability is more important than vertical leap.

"Barefoot Shoes" have thin elastic soles to bring out tactile contact with the ground, but they're not sturdy.

Shaolin kung fu

The Shaolin temple
A tree in the Shaolin temple

Mind-body hierarchy

Shaolin kung fu is the highest art. All other arts are subsets of Shaolin.

Tier 1:  Shaolin, Wushu, Sumo
Tier 2:  Feldenkrais Method
Tier 3:  Alexander Technique,  Reiki
Tier 4:  Physical therapy

Graphic novels

If you want superpowers, learn physics.

Iron Man        Physicist and mechanical engineer
Magneto         Physicist.  Mutant.  Control over magnetic fields.
The Hulk        Physicist.  Irradiated with gamma rays
Dr. Manhattan   Physicist.  Atomized in an "intrinsic field subtractor".
Bruce Wayne     Physicist and mechanical engineer.
The Beast       Physicist.  Mutant.
Professor X     Physicist.  Mutant.  Telepath.
The Owl         Physicist and mechanical engineer.
Ozzymandias     Physicist.  Master of Shaolin kung fu.
Howard Stark    Physicist and mechanical engineer
Jarvis          AI created by Tony Stark.
The Vision      AI created from Jarvis.  Possesses the mindstone.  Vibranium body.
The Shadow      Shaolin monk
Dr. Selvig      Astrophysicist
Jane Selvig     Astrophysicist
Silver Surfer   Astrophysicist.  Possesses the silver surfboard.
Mr. Fantastic   Physicist.  Acquired superpowers from cosmic rays.
The Thing       Engineer.  Fighter pilot.  Football player.
                Acquired superpowers from cosmic rays

1930-      The Shadow
1938-1950  Golden Age of Comic Books
1939-      Batman
1956-1970  Silver Age of Comic Books
1963       X-Men
1970-1985  Bronze Age of Comic Books
1985-      Modern Age of Comic Books
1986       Watchmen 


Replaced skeleton with adamantium
Intrinsic field subtractor
Always angry
Trained at a Tibetan temple

If you want superpowers, you have to develop your pain tolerance.

Muggle fu

If you're at level 1, there are fast and easy techniques that will get you to level 2, but they come with cheats that will block you from ever reaching level 3. If you want to be a wizard, you have to train fundamentals. The depth of your fundamentals determines your ultimate potential.

The only way to properly test technique is against worthy competition.

Alexander Technique

F. M. Alexander

Alexander Technique emphasizes gaining awareness of the atlas and axis vertebra.

American Society for the Alexander Technique (AmSAT).
Alexander Technique International (ATI).
Society of Teachers of the Alexander Technique (STAT).

Localized joint releases, especially of the atlas-axis vertebra.

Hand stability for projecting precise forces.

Emphasizes the motion of standing up from a chair.

Stabilize torque noise in the limbs.

Emphasize the "means-whereby" as opposed to "end-gaining".

Think ahead: Simulate in your mind the upcoming sequence of events before they occur.

Lengthen the spine.

Mantra: "Head forward and up"

Bodymapping: Awareness of the pivot points of joints.

The Feldenkrais Method

Moshe Feldenkrais
Moshe Feldenkrais

Feldenkrais was a physicist and wrestler.

Awareness through movement: Move in slow-motion and isolate the motion of joints.

Eye training: Move the eyes in slow motion and train going back and forth between near-focus and far-focus.

Move according to the normal modes of the skeleton.

Societies: The Feldenkrais Guild of North America.

Craniosacral therapy

A craniosacral therapist works on the relationship beween the two ends of the spine.


Reiki uses an awareness of pressure fluctuations in the body to quell noise.

Tai Chi


Darth Maul
Ray Park

Darth Maul was portrayed by Ray Park, a wushu master.

Jet Li won the Chinese wushu championship at age 12.

Bruce Lee's Jeet Kune Do

Bruce Lee: "Jeet Kune Do favors formlessness so that it can assume all forms and since Jeet Kune Do has no style, it can fit in with all styles. As a result, Jeet Kune Do utilizes all ways and is bound by none and, likewise, uses any techniques which serve its end."

Bruce Lee: "Stylists, instead of looking directly into the fact, cling to forms (theories) and go on entangling themselves further and further, finally putting themselves into an inextricable snare."

Bruce Lee: "Truth has no path. Truth is living and, therefore, changing. Awareness is without choice, without demand, without anxiety; in that state of mind, there is perception. To know oneself is to study oneself in action with another person. Awareness has no frontier; it is giving of your whole being, without exclusion."


Bruce Lee: "All vague notions must fall before a pupil can call himself a master."



The Cray-1, developed in 1976
The Kraken Supercomputer, built in 2009

c. -2400  Abacus invented by the Babylonians
c.  -500  First known use of the number "0" in Ancient India
c.  -300  Pingala develops binary numbers
c.  -100  Negative numbers used in Ancient China
c.   200  Logarithms developed in Ancient India
c.   600  Brahmagupta develops a place-value number system
c.  1400  Kerala school of astronomy and mathematics in South India invents the
          floating point number system.
    1585  Stevin popularizes decimal numbers in Europe
    1614  Napier develops logarithm tables
    1622  Oughtred develops the slide rule
    1642  Pascal builds a mechanical calculator
    1671  Leibniz builds a mechanical calculator
    1725  Bouchon and Falcon use punch cards to program looms
    1910  Vacuum tube invented
    1646  ENIAC built.  17468 vacuum tubes.  5000 additions per second.
    1947  Transistor invented
    1957  Fortran compiler developed. This was the first high-level programming language
    1967  Dijkstra declares that the "goto" statement is harmful and advocates
          structured programming

Fastest supercomputer as a function of time

The world's fastest supercomputer is the Tianhe-2, built in 2013. It has the following properties:

Speed                  34 Petaflops
Processor cores    384000
Memory               1357 Terabytes
Disk drives          12.4 Petabytes
Power                  24 MWatts
Footprint             700 meters^2
Punch card with the statement "Z(1) = Y + W(1)"
64 GB of punch cards
64 GB microSD card
Jacquard loom, invented in 1801

Many-body theory

2-body problem
3 bodies evolving chaotically

The "Kepler" problem consists of two gravitationally-orbiting objects and it was solved by Newton in the Principia Mathematica in 1687. This is an example of a "two-body problem". An example of a "three-body problem" is the sun, the Earth, and the moon. In the 18th century there was much interest in solving this problem because a solution would provide a means for determining longitude at sea, but it was realized that no simple closed-form solution is possible. In 1887 Poincare discovered the phenomenon of chaos and he found that the three-body problem can behave chaotically, meaning that is long-term behavior is impossible to predict.

"It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the many-body problem." - Max Born, 1960

Heisenberg: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first."

Molecules in a gas
Brownian motion

The ideal gas equation is an example of a simplification of a many-body problem.



Leopold Auer: There are those who in specific monographs have advised 'relaxation' of the hand on every occasion! Well, I myself believe in relaxation, when the word denotes general repose when working, or is employed as a synonym for a certain elasticity of the bow-arm, freedom of the wrist, and light pressure of the fingers on the stick. But, if we are speaking of 'relaxation' of hte left hand, that is to say, of the fingers of the left hand, then I am of the contrary opinion. In fact, I believe so strongly that this is a false conception, that I feel I must insist that the pressure of the fingers must conform in exact measure to their physical strength. I will even go so ffar as to say that the more one tries to diminish the body of tone, in a piano and pianissimo, for instance, the more one should increase the finger pressure, especially in the positions where the strings are raised higher above the fingerboard, and in the acute notes on the E-string. The greater the finger pressure you exert in this region of the fingerboard, the more rapidly these notes will vibrate beneath a slight pressure of the bow.

Adaptive resolution

Smoothed Particle Hydrodynamics (SPH)
Adaptive Mesh Refinement (AMR)

In a simulation it can be useful to be able to focus resolution on regions of interest. The above figure shows how this is done for a grid and particle algorithm.

The brain performs adaptive resolution when needed and when the resolution is no longer needed the brain dismantles the extra resolution.

Ground state

Electron descending an energy level. The ground state is n=1.

In the above left figure, the particle above is not in the ground state for energy. To be in the ground state the particle has to be stationary at the bottom of the valley.

Global optimization

The midwest
Bryce canyon
Local maximum

The goal of global optimization is to find the maximum or minimum of a multidimensional function. If the function is smooth, such as the midwest, finding the minimum is easy. If the function is rugged, like Bryce canyon, finding the minimum is difficult.

In the above right panel the red point is the maximum of a 2-dimensional smooth surface.

Steepest descent, easy case
Steepest descent, difficult case
Steepest descent, extremely difficult case

In the "steepest descent" method one proceeds in the direction of the local downhill. This method cannot distinguish between a local or global minimum and it doesn't work well if the valleys are narrow.

In global optimization you have to resist overfocusing on any single variable.


If you cool a metal slowly it finds the state of minimum-stress. A metal is annealed by repeatedly heating and cooling it with slow changes in temperature.

Siulated annealing is a method for avoiding local minima and finding the global minimum.

Fast cooling
Slow cooling

If you cool a metal slowly it finds the state of minimum-stress. A metal is annealed by repeatedly heating and cooling it with slow changes in temperature.

Genetic algorithms

History of mass extinctions

Self similarity

Practice playing at different speeds and practice going back and forth between light and heavy forces.

Critically-damped harmonic oscillator

q  =  Quality parameter of an oscillator
The damping is inversely proportional to the quality parameter. The behavior of the oscillator as a function of q is shown in the figure above. The colored curves correspond to:
Red:      q <  1/(2Pi)       Overdamped
Cyan:     q =  1/(2Pi)       Critically-damped
Purple:   q >  1/(2Pi)       Underdamped
Yellow:   q =  Infinity      Not damped
If the oscillator is underdamped then it will pass through zero before damping out. q=1/(2Pi) is the boundary for which the oscillator doesn't cross the axis before damping. This is "critical damping".

Your muscles should function like a critically-damped harmonic oscillator, dissipating energy while generating a minimum of noise.

When the goalkeeper stops the ball, if the force is too small the ball will go through his hand and into the goal. If the force is too large the ball will rebound back into play. The stopping force has to be tuned to bring the ball to a halt without rebounding.


Gaussian curves
Convolution blurring
2D Gaussian

Differential equations

Black: no drag.    Blue: Stokes drag.    Green: Newtonian drag
Golf trajectory

Euler method
Red: Euler method.    Green: midpoint method
Same as to the left but with smaller timesteps

The important chapter in the Numerical Recipes textbook is "Integration of ordinary differential equations".

Gamma function
Chebyshev polynomials
Chebyshev polynomials


In the first stage of a predictor-corrector timestep, one "predicts" the forward time evolution from the present conditions. In the second stage one "corrects" the trajectory using information from the future timestep.


Spline interpolation
Polynomial interpolation

The red points are data and the black curve is a cubic spline interpolation.

Never use a polynomial for interpolation or extrapolation. The polynomial can go on wild swings between points.

Computational fluid dynamics

Computational fluid dynamics is the art of simulating fluids on a computer. It represents the state of the art of techniques for evolving differetial equations and it is rich in lore that is essential for understanding the mindbody system.
* The Phurba algorithm: Theory.    "Phurbas: An Adaptive, Lagrangian, Meshless, Magnetohydrodynamics Code. I. Algorithm" Maron, McNally & Mac Low, ApJS 2012, Vol 200, 1, 6
* The Phurba algorithm: Tests.    "Phurbas: An Adaptive, Lagrangian, Meshless, Magnetohydrodynamics Code. II. Implementation and Tests" McNally, Maron & Mac Low. ApJS 2012, Vol 200, 1, 7


Turbulence is described using the mathematics of cascades, and this mathematics is essential for understanding muscle noise.

* Magnetized turbulence.    "The Nonlinear Magnetic Cascade," Maron, Cowley, & McWilliams, ApJ 603, 2004
* Helical dynamo.    "Effect of Fractional Kinetic Helicity on turbulent Magnetic Dynamo Spectra" Maron & Blackman, ApJL 566, 41, 2002
* Alfven wave turbulence "Simulations of Incompressible MHD turbulence" J. Maron & P. Goldreich, ApJ 2001

Zone defense

Zone defense
Man-to-man defense

There is a set of parallel eye circuits that activate depending on the direction and orientation of the source object.


Massively-parallel supercomputer
Graphical processing unit
Gatling gun

A graphical processing unit (GPU) has a set of floating point CPUs that function in parallel, usually around 256 of them. All CPUs have to execute the same instruction sequence.

The brain is good at keeping track of multiple processes at once, and it can do it subconsciously without you needing to be aware of it.

Calculations that can be massively parallized are: Interpolation, differential equations, convolutions.

Peripheral vision and peripheral hearing

Corrective lenses are an obstacle to developing peripheral vision.

Monte Carlo integration

You can evaluate the area of the circle by throwing random darts at the plane and counting the fraction that land in the circle.

If an integral has a complex boundary then Monte Carlo integration provides a way to sidestep the complexity.

Random numbers

Lorentz attractor
Poisson attractor
Radioactive decay

It's easy to design a random number generator that is uniform in 1 dimension and difficult to design one that is uniform in higher dimensions.


In chaotic systems, nearby initial conditions can evolve to divergent states. This makes it impossible to predict weather more than a week in advance.

Green's function

Dirac delta function
Plucked at the center of string
Plucked at the edge of the string

A Dirac delta function is the limit of an infinitessimally-thin Gaussing, with the area under the curve preserved in the limit.

If the string is plucked at any given point then the motion can be solved exactly.

Suppose you solve the shape of a wire for a bird standing at any possible place on the wire. If there are multiple birds then the shape of the wire is the sum of the shapes from each individual bird.

If the string is bent into an arbitrary shape it can be decomposed into a sum of Dirac delta functions and the solution is the sum of the delta functions.

Floating point operations

The following table gives typical times for common floating point operations with respect to the clock cycle.

A typical clock time is 10^{-10} seconds, which corresponds to a frequency of 10 GigaHertz.


Clock cycle       1
Add               3
Subtract          3
Multiply          3
Divide           12
Exp              24
Sin              24
X^Y              32
If                3
Absolute value    3
Type conversion   3
Register access   1         Typically 1 KB in size
L1 Cache          3         Typically 64 KB in size
L2 Cache         16         Typically 256 KB in size
L3 Cache         64         Typically 8 MB in size
RAM             256         Typically 8 GB in size
Adjacent CPU   1000
Distant CPU   10000
Hard drive   100000
SD card     1000000

CPUs tend to be data-starved in that floating point arithmetic is faster than memory access. The brain tends to be data-swamped because the synapse time is longer than the time it takes a neural signal to cross the brain.

Floating point pipeline

Functioning pipeline
Stalled pipeline

Helical noise

Magnetic field dynamo
Earth's magnetic field
Solar magnetosphere

Magnetized turbulence tends to resolve longitudinal noise faster than helical noise, and the same is true for the normal modes of your skeleton. The noise in your skeleton is dominated by helical noise.


Algorithms based on matrices are usually not transparent nor are they failsafe. Avoid them.


Order of magnitude physics

Order of magnitude style (Fermi style) is battle physics, the art of calculating numbers when the input data is imprecise.

Once upon a time, Peter Goldreich and Sterl Phinney pioneered an order-of-magnitude physics class at Caltech.

"The Art of Insight in Science and Engineering", Sanjoy Mahajan
"Street-Fighting Mathematics, The Art of Educated Guessing and Opportunistic Problem Solving", Sanjoy Mahajan
"Order-of-Magnitude Physics: Understanding the World with Dimensional Analysis, Educated Guesswork, and White Lies", Peter Goldreich, Sanjoy Mahajan, and Sterl Phinney
"Order-of-Magnitude Physics", Jay Maron
"The Solar System", Eric Blackman
"Astropedia", Chris Impey
"The astronomical reach of fundamental physics", Adam Burrows and Jeremiah Ostriker
David Hogg, "Real-world physics: A dropped bucket"
David Hogg, "Air resistance"
Sanjoy Mahanan and David Hogg, "Introductory Physics, the new Scholasticism"
Sterl Phinney, Caltech
Eugene Chiang, Berkeley
Nir Shaviv, Hebrew University of Jerusalem

Youtube videos
XKCD articles
Minute Physics & Minute Earth


The Android App "FrequenSee" visualizes the frequency spectrum of sound.

The iPhone App "Garage Band" functions as a musical instrument and also can visualize the frequency spectrum of sound.

Seymour Cray

When asked what kind of CAD tools he used for the Cray-1, Cray said that he liked #3 pencils with quad paper pads. Cray recommended using the backs of the pages so that the lines were not so dominant. When he was told that Apple Computer had just bought a Cray to help design the next Apple Macintosh, Cray commented that he had just bought a Macintosh to design the next Cray.


Flying cars


The principal obstacle to flying cars is noise. There is no way to fly quietly. To fly you have to move a lot of air.

The larger the propeller, the less noise that is generated for a given lift.

                     Energy/Mass    Energy/$     Recharge     Max       Power
                     (MJoules/kg)   (MJoules/$)   time       charges    density
Diesel fuel             47          41.             -           -          1
Lithium-ion battery       .95         .009        10 hours     1000        1
SuperCapacitor            .036        .001        Seconds     Millions    10

Electric cars

The Tesla Roadster

Suppose an electric car is such that
M  =  Mass                  = 1000 kg
A  =  Cross-sectional area  =    3 m^2
D  =  Density of air        =  1.2 kg/m^3
V  =  Velocity
E  =  Kinetic energy
F  =  Drag force
   =  .5 D A V^2
P  =  Drag power
   =  .5 D A V^3
X  =  Distance traveled
Ed =  Drag energy from moving a distance X
   =  F X
Distance traveled such that the kinetic energy is equal to the drag energy:
.5 D A V^2 X  =  .5 M V^2

X  =  M / (D A)
   =  280 meters
Energy of a 1 ton car moving at 15 m/s = .11 MJoules. This can be stored with 4 kg of capactors. Capacitors are ideal for absorbing breaking energy and for city driving.

Energy required to travel 10 km at 15 m/s = 4.0 MJoules

Power required to travel at 15 meters/second = 6 kWatts

Inductive chargers have been built that can handle a power of 50 kWatts.

100 kg of capacitors costs $3600, has an energy of 3.6 MJoules, and can power a car moving at 15 m/s for 9 km. If a city has abundant charging stations then capacitor-based cars can work in cities. If this car is moving on a freeway at 30 m/s then it can cover 2.2 km. Capacitor-based cars are not useful for traveling between cities.

Energy efficiency of a vehicle

D  =  Air density
A  =  Cross-sectional area
V  =  Velocity
F  =  Drag force
   =  .5 D A V^2
P  =  Drag power
   =  .5 D A V^3
X  =  Distance traveled
E  =  Drag energy
   =  F X
N  =  Number of people in the vehicle
Z  =  Energy efficiency
   =  Drag force per person
   =  F / N

Drag force is equal to Energy/Distance. The measure of the energy efficiency of a vehicle is Z = Force/Person. In the following table, we calculate the force/Person using typical values for each vehicle.

                    Speed  Area  Air      People   Force   Force/     Power
                     m/s    m^2  density            kN     Person      kW
                               (kg/m^3)                  (kN/person)
City car             15     3     1.2         1      .4     .4          6.1
City bus             15    10     1.5        20     1.4     .07        20
Freeway car          30     3     1.2         1     1.6    1.6         49
Freeway bus          30    10     1.2        60     5.4     .09       162
Train                50    20     1.2      1000    15       .015      750
Passenger airplane  300    40      .2       200   360      1.8     108000
F-22 Raptor         670            .2         1   312    312

Buses and trains are better than airplanes and cars. If you put 4 people in a car it is still less efficient than a bus or train.

Sound generation

The motion of a fluid can be decomposed into monopoles, dipoles, and quadrupoles. A monopole is like an explosion or implosion, a dipole is like a jet, and a quadrupole is like subsonic isotropic turbulence. The loudness of sound produced depends on the type of motion.

M  =  Mach number of the turbulence
   =  Characteristic speed of turbulent fluctuations / Speed of sound

Monopole:     Loudness scales as M
Dipole:       Loudness scales as M^2
Quadrupole:   Loudness scales as M^3

A fluid flow can be engineered to avoid dipole structure and minimize sound generation. The fluid flow can also be simulated to find a solution that minimizes the generation of turbulence.

Turbulence generates sound wave energy. The sound wave energy is generally substantially less than the turbulent energy and so the sound waves can be neglected. Turbulence is transparent to the sound waves it generates. In an anechoic chamber, the walls are made out of cones designed to not reflect sound back into the room.

The ear

The timpanic membrane converts air pressure waves into mechanical motion of the ear bones. The ear bones amplify the signal and transmit it to the stapes bone, which is connected to the oval window of the cochlea. Vibrations in the oval window cause vibrations in the fluid of the cochlea, where they are converted into neural signals and interpreted in the brain.

If a sound wave in air encounters water then 1/30 of the sound energy is transmitted to the water and the rest reflects back into the air. If sound waves were transmitted directly from the air to the fluid of the cochlea then they would suffer this loss. The ear bones function to improve the transmission efficiency from the air to the fluid of the cochlea.

The tympanic membrane has 13 times the area of the oval window, and so the signal is amplified by a factor of around 13.

As pressure waves travel along the cochlea the cochlea narrows. The narrower the cochlea, the higher the frequency range it is sensitive to. Low frequencies are detected at the beginning of the cochlea and high frequencies are detected at the end.

If the sound level is too loud then the muscles of the middle ear shut down the motion of the ear bones. This is the "acoustic reflex" (Wikipedia).

The function of the ear bones was first explained by Helmholtz.

The cochlea


A microphone records sound pressure as a function of time and a seismometer records displacement as a function of time. Your ears don't work anything like this. Your ears function instead detect the frequency spectrum, analogous to a spectrum of light.

Cross-section of the cochlea showing the organ of Corti

There are 20000 hairs arranged along the length of the cochlea, each tuned to a different frequency. Each hair functions as a resonator.

High frequencies are detected at the start of the cochlea and low frequencies are detected

The perceived loudness depends on the duration of the note. For notes less than .2 seconds the loudness is proportional to duration and for notes more than .2 seconds the loudness is independent of duration. This suggests that the cochlea functions like a resonator, because it takes time for a resonance to activate.

If the duration of a note is much longer than 1 second then our attention fades and the note seems less loud.

Our ability to resolve frequency depends on the sharpness of the resonators in the cochlea. The brain provides active feedback to sharpen the resonance and suppress resonances at nearby frequencies.

If there are two sounds with different frequencies, then if the frequencies are too close to each other they will interfere with each other in the cochlea, and if they are far enough apart they can be sensed independently. The frequency width for interference is on the order of a perfect fifth.

Noise tend to obstruct our ability to resolve pitch.

Nerve signals travel both from the cochlea to the brain and from the brain to the cochlea. The brain provides active feedback to refine the function of the cochlea.

There are nerves that travel directly back and forth between your ears for stereo processing.

The semicircular canals of the cochlea are a gyroscope. Rotating your head causes fluid to flow in the canals, which is detected by hair cells. The function of the gyroscope and the function of the auditory system are connected.

The vestibule and the saccule are hardened objects used to detect linear acceleration. WHen you accelerate these objects are displaced, which is detected by hair cells.

Your ears are at the center of your skull, aligned with the pivot point that connects your skull to the top of your spine. The ear is involved in calculating balance.

Basilar membrane

The basilar membrane functions like a harp or piano. It is a strip running the length of the cochlea, narrow at the end closest to the ear and wide at the end farthest from the ear, like a necktie. It is also stiffer the closer it is to the narrow end. The resonant frequency at any particular point along the basilar membrane increases with stiffness and decreases with width, giving it a frequency range that varies from high to low as you traverse from the narrow to the wide end. Siffness is controlled with muscle tension.

The lower the frequency of the wave, the further it propagates along the basilar membrane. High-frequency waves diminish before they get to the wide end.

The fact that low frequencies propagate further along the basilar membrane is analogous to the fact that low-frequency pitches more easily pass through walls than high-frequency pitches. A low-frequency pitch has more time to move the wall for a given sound pressure.

Youtube: Basilar membrane

Helmholtz was the first to characterize the function of the basilar membrane

Absolute pitch

One out of every 10000 people has "absolute pitch", where for example you can tell if a pitch is higher, lower, or equal to 440 Hertz. Everone else has "relative pitch", where pitch ratios can be sensed but not absolute pitch. This suggests that there is no fixed place on the basilar membrane that corresponds to 440 Hertz.

If you don't have absolute pitch it is difficult to acquire it. From Wikipedia: "There are no reported cases of an adult obtaining absolute pitch ability through musical training; adults who possess relative pitch, but who do not already have absolute pitch, can learn pseudo-absolute pitch, and become able to identify notes in a way that superficially resembles absolute pitch. Moreover, training pseudo-absolute pitch requires considerable motivation, time, and effort, and learning is not retained without constant practice and reinforcement."

Frequency and time

The ear is sensitive to both pitch and time. Pitch is measured by position along the basilar membrane and time is measured by differences beween neural signals. For high-frequency pitches we are more sensitive to frequency and for low-frequency pitches we are more sensitive to time.


Fennec fox

The pinna is the outer part of the ear that collects sound and helps in determing its directionality. All human pinna are unique in shape and if the shape were to change it would affect your ability to determine the direction of sound.

A large pinna can amplify sound by 10 to 15 decibels.

Haas effect for echos

Suppose a sound pulse arrives at your ear and an echo arrives a time T later. If T < 30 ms then you don't notice the echo and if T > 30 ms you notice the echo.

The distance a sound wave travels in a time of 30 ms is 10 meters. A concert hall has to be smaller than this to not sound like it has echos.

To do echolocation you have to train your ears to be sensitive to intervals less than 30 ms.

Bats use high frequencies for echolocation because they diffract less than low frequencies and hence give better resolution.

Ear training

Anguy   Keep your elbow high.  You want your back doing the hard labor.
        You're holding.  Never hold.
Arya    What?
Anguy   Your muscles tense up when you hold.  Pull the string back to the center
        of your chin and release.  Never hold.
Arya    But I have to aim.
Anguy   Never aim.
Arya    Never aim?
Anguy   Your eye knows where it wants the arrow to go.  Trust your eye.

Speed of sound

Bruce Lee: Experiments indicate that auditory cues, when occurring close to the athlete, are responded to more quickly than visual ones. Make use of auditory clues together with visual clues, if possible. Remember, however, the focus of attention on general movement produces faster action than focus on hearing or seeing the cue.

Bruce Lee: You hear the bird chirping? If you don't hear the bird you cannot hear your opponent.

Neuron             100
Sound in air       343         At 20 Celsius
Sound in water    1482
Light        300000000

Time in milliseconds:

.000003  Time for light to cross a 10 meter orchestra
    .2   Electric synapse. These synapses are 2-way and they do not amplify signals
    .7   Time for a water pressure wave to travel 1 meter through your body
   2     Chemical synapse. These synapses are 1-way and they can amplify signals
   1     Time for a neural signal to travel 10 cm, the size of a brain
  10     Time for a neural signal to travel from your fingers to your brain
   3     Time for sound to travel 1 meter, the distance to an adjacent musician
   7     Period of a 130 Hertz wave. This is the frequency of a viola C string
  30     Time for sound to travel across a 10 meter orchestra.
  62     Time between notes in "Flight of the Bumblebee"
For an orchestra to have good timing it must use visual cues. Sound isn't fast enough. This is especially true at the rear of the viola section amidst the cacophony of winds and brass.

The Europa Galante uses precise visual cues.

Pressure waves in your muscles deliver information 15 times faster than neurons.

Listen down

When listening to an orchestra one's attention most easily falls on the high-frequency instruments. Practice listening to the low-frequency instruments, especially the cellos and basses. They control the long-term temporal coherence.

Bruce Lee: You hear the bird chirping? If you don't hear the bird you cannot hear your opponent.

If you can't hear the violas you can't hear the chord. Practice listening to the middle note of the chord.

One should also practicing listening to instruments at minimal volume. Loud volume obstructs our ability to resolve pitch.

Listen to silence

The lower the sound intensity, the more sensitive we are to pitch. Practice listening to music at minimal volume.

Listen ahead

Anticipate the pitch in your mind before you play it. The cochlea has active feedback from the brain and this helps harness it.

Develop fast reactions to adjust the pitch to be in tune with the rest of the orchestra.

Frequency sensitivity of the human ear

Frequency   Wavelength
 (Hertz)     (meters)

   20        15        Lower limit of human frequency sensitivity
   41         8.3      Lowest-frequency string on a string bass or bass guitar
   65         2.52     Lowest-frequency string on a cello
  131         2.52     Lowest-frequency string on a viola
  440          .75     The A-string on a violin
  660          .75     The E-string on a violin (highest-frequency string)
20000          .016    Upper limit of human hearing



Vestibular system

Semicircular canals
Semicircular canal

The semicircular canals are .8 mm in diameter.

Vestibulo-Ocular reflex

The eyes detect head movement from the vestibular system and use it to stabilize the image.

Eye Saccades

The eye moves in sudden jumps. It will be stable for an interval and then it will make a discontinuous jump before returning to stability. The jumps are called "saccades". Saccades are analogous to Earthquakes. When the eye is held stable tension builds up until a saccade occurs.

Youtube: Eye saccades in slow motion


Visual information crosses at the optic chiasm before being assembled at the rear of the brain.

The motor cortex is in front of the somatic cortex.

Optic chiasm

Information from the eyes crosses at the optic chiasm.


Corpus callosum

The corpus callosum connects the two brain hemispheres. It is tangibly larger and more plastic in musicians.

Endocrine glands
Pineal gland

Energy dissipation

Power at maximum exertion       = 1500 Watts
Power used by the body at rest  =  100 Watts
Power used by the brain         =   20 Watts

Motor cortex

Motor cortex
Motor cortex
Motor and somatic cortex

The motor cortex is in front of the somatic cortex.

Information from the eyes passes through the motor cortex before being assembled at the rear of the brain. The motor cortex is an image-stabilization system for the eyes. Visual input requires neural processing before it can be interpreted.

In the motor cortex, proceeding from the center to the edge of the brain corresponds to proceeding from the feet to the head of the body. It represents a stack of reference frames starting from the ground and proceeding upward.

Cerebrospinal fluid

Composition of the brain
Distribution of cerebrospinal fluid

The brain produces 500 mL of cerebrospinal fluid per day and at any given time there is 100-160 mL present.


Cell membrane

Cell membranes assemble spontaneously from phospholipid molecules. They are mechanically flexibible due ot the ability of the phospholipids to rearrange themselves.

Cell membranes pass fat-soluble molecules and block water-soluble molecules. Proteins can move molecules across the membrane.

Ion pump

A membrane has ion channels that passively permit ion flow, and ion pumps that actively transport ions. Most ion channels are permeable only to specific types of ions. Ion channels can be modulated by either the membrane voltage or by chemicals.

The sodium-potassium pump generates a membrane voltage of around 70 mVolts, with the cell interior being negative.

In each cycle of the sodium-potassium pump, 3 sodium ions move outward and 2 potassium ions move inward. The pump requires hours to establish equilibrium. The pump is powered by ATP and the voltage gradient it produces provides a power source for other ion pumps.

In each cycle of the sodium-calcium pump, 3 sodium ions move inward and 1 calcium ion moves outward. This pump is powered by the membrane potential and doesn't require ATP.

An ion channel for a given ion does not pass larger ions, and most ion channels are specific for one ion. For example, most potassium channels are characterized by 1000:1 selectivity ratio for potassium over sodium, though potassium and sodium ions have the same charge and differ only slightly in radius. The pore is small enough so that ions must pass in single-file.

An action potential involves the opening and closing of ion channels and doesn't involve ion pumps. If the ion pumps are turned off by removing their energy source, or by adding an inhibitor such as ouabain, an axon can still fire hundreds of thousands of action potentials before the amplitudes begin to decay significantly.

The chloride ion is not actively pumped and takes on an equilibrium concentration given by the membrane potential.

Potassium channel protein with a potassium ion in the center

Resting potential
Skeletal muscle cells  = -95 mV
Smooth muscle cells    = -60 mV
Astroglia (Glia cells) = -85 mV +- 5 mV
Neurons                = -65 mV +- 5 mV
Red blood cells        =  -8 mV
Photoreceptor cells    = -40 mV


Microscope image

Signals travel from the cell body outward along an axon, jump to the dendrite of another neuron at a synapse, then travel inward along the dendrite to the cell body of the new neuron.

A neuron has at most one axon but the axon can branch hundreds of times. A neuron can have multiple dendrites. There are, however, many exceptions to these rules: neurons that lack dendrites, neurons that have no axon, synapses that connect an axon to another axon or a dendrite to another dendrite, etc. In certain sensory neurons (pseudounipolar neurons), such as those for touch and warmth, the electrical impulse travels along an axon from the periphery to the cell body, and from the cell body to the spinal cord along another branch of the same axon.

The longest axons in the human body are those of the sciatic nerve, which run from the base of the spinal cord to the big toe of each foot. The diameter of axons is also variable. Most individual axons are microscopic in diameter (typically about one micrometer across)

Brain neurons               = 100   billion
Brain neurons (cerebrum)    =  16.3 billion
Brain neurons (cerebellum)  =  69   billion
Brian glia cells            = 100   billion
Brain synapes               = 100   trillion
Neuron volume / Glia volume =   1.0

Neuron speed (with myelin)  = 100   m/s
Neuron speed (no myelin)    =   2   m/s
Axons for motor muscles     = 100   m/s            (16 um diameter)
Axons for sensory muscles   =  10   m/s            ( 8 um diameter)

Size of brain               =  15      cm       =  15000 neurons across
Distance between neurons    =  10      μm
Axon diameter (large)       =  20      μm
Axon diameter (small)       =   1      μm
Membrane thickness          =    .0075 μm
Chemical synapse gap        =    .020  μm
Electric synapse gap        =    .0035 μm
Node of Ranvier diameter    =   1.5    μm +- .5 μm
Node of Ranvier spacing     =1000      μm        (Distance between adjacent nodes)
Axon max size in humans     = 106      μm
Dendrite max size in humans =1000      μm
Electric synapse diameter   =    .0016 μm
Electric synapse length     =    .0075 μm

Neuron body ion channels    =   1      μm-2
Axon hillock ion channels   = 150      μm-2
Myelin ion channels         =  25      μm-2
Node of Ranvier ion channels=5000      μm-2   (Between 2000 and 12000 μm-2)

Brain neuron density        =    .0010 μm-3
Brain synapse density       =   1.0    μm-3

Chemical synapse time       =   2.0    ms
Electric synapse time       =    .2    ms
Sodium action potential     =   1      ms      (Duration)
Calcium action potential    = 100      ms      (Duration)
Sodium-Potassium pump time  = 107      ms      (Hours)  (Time to reach equilibrium)

Spines per dendrite         =1000
Sodium ratio                =   9        (Exterior concentration / interior concentration)
Potassium ratio             =  20        (Interior concentration / exterior concentration)
K+ current / Na+ current    =  20        (Current across membrane in resting state)

Typical membrane potential  =   -.07 Volts     (The cell interior is negative)
Sodium reversal potential   =   +.10 Volts
Potassium reversal potential=   -.90 Volts
Chloride reversal potential =   -.07 Volts     (Same as resting potential)
Membrane breakdown voltage  =    .2  Volts
Breakdown field (air)       =   3    Volts/μm
Breakdown field (membrane)  =  27    Volts/μm
Breakdown field (vacuum)    =  30    Volts/μm
Breakdown field (water)     =  68    Volts/μm
Membrane capacitance        =   2    uF/cm2

Max action potential rate   = 100    seconds-1

Axon diam. / Nerve diam.    =    .6        Nerve diameter corresponds to axon plus myelin sheath


1)  Unipolar neuron. Axon and dendrite emerging from same process.
2)  Bipolar neuron. Axon and dendrite on opposite ends.
3)  Multipolar neuron. One axon and many dendrites.
4)  Anaxonic. The axon can't be distingished from the dendrites.

Axons connect to the cell body through the axon hillock. The axon hillock is the last site in the cell where membrane potentials propagated from synaptic inputs are summated before being transmitted to the axon.

Action potential

If the voltage across the membrane exceeds the threshold, voltage-gated sodium ion channels open and sodium rushes into the cell, accelerating the voltage rise. When the voltage reaches its peak the sodium channels close and potassium channels open, restoring the potential to its resting state.

If the voltage change is too small to cross the threshold, the potassium current exceeds the sodium current and the voltage returns to its normal resting value.

After the action potential fires the axon enters a refractory state, which is responsible for the unidirectional propagation of action potentials along axons. At any given moment, the patch of axon behind the actively spiking part is refractory, but the patch in front, not having been activated recently, is capable of being stimulated by the depolarization from the action potential.


A myelin coating increases the speed of signals. Myelinated axons are known as nerve fibers.

Signal propatation in myelinated axons is called "saltatory conduction", where the signal jumps rapidly from one node of Ranvier to the next.

In the central nervous system (CNS), myelin is produced by oligodendroglia cells. Schwann cells form myelin in the peripheral nervous system (PNS). Schwann cells can also make a thin covering for an axon which does not consist of myelin (in the PNS). A peripheral nerve fiber consists of an axon, myelin sheath, Schwann cells and its endoneurium. There are no endoneurium and Schwann cells in the central nervous system.

In myelinated axons, ionic currents are confined to the nodes of Ranvier and far fewer ions leak across the membrane than in unmyelinated axons, saving metabolic energy.

Myelin decreases capacitance and increases electrical resistance across the cell membrane.

Myelin permits large organisms to exist by enabling fast communication between distant body parts.

When a peripheral nerve fiber is severed, the myelin sheath provides a track along which regrowth can occur. However, the myelin layer does not ensure a perfect regeneration of the nerve fiber. Some regenerated nerve fibers do not find the correct muscle fibers, and some damaged motor neurons of the peripheral nervous system die without regrowth. Unmyelinated fibers and myelinated axons of the mammalian central nervous system do not regenerate.

Chemical synapse

When an axon signal reaches a synapse, calcium channels open and calcium flows into the cell. Vesicles, which store neutrotransmitters, then open and release neurotransmitters into the synaptic gap. The neurotransmitters diffuse across the synaptic gap and bind to the target cell, triggering an action potential in the target cell.

Synapses are usually located at the terminals of axons but they can also be located at junctions along the axon ("in passing" synapses). A single axon with all its branches can innervate multiple parts of the brain and generate thousands of synaptic terminals.

A chemical synapse can amplify signals and an electric synapse cannot.

                   Time    Spacing
                   (ms)     (nm)
Chemical synapse    2        30
Electric synapse     .2       3.5
Lipid vesicle
Neurotransmitters are stored in lipid vesicles.

Lipid vesicles

Electric synapse

Electric synapses are faster than chenical synapses but they can't amply signals like chemical synapses.

In an electric synapse, signals jump from the membrane of one cell to another through a connexon joint. A connexon joint is a tunable iris composed of 6 connexin proteins.

The response is always the same sign as the source. For example, depolarization of the pre-synaptic membrane always induces depolarization in the post-synaptic membrane, and vice versa for hyperpolarization.

The response in the postsynaptic neuron is in general smaller in amplitude than the source. The amount of attenuation of the signal is due to the membrane resistance of the presynaptic and postsynaptic neurons.

Because electrical synapses do not involve neurotransmitters, electrical neurotransmission is less flexible than chemical neurotransmission.

Long-term changes can be seen in electrical synapses. For example, changes in electrical synapses in the retina are seen during light and dark adaptations of the retina.

Glial cells

Astrocytes in blue

Astrocytes in blue
Astrocytes in blue

Astrocytes          Provide nutrients to neurons
Microglial cell     Cleanup
Oligodendrocyte     Add myelin to axons in the central nervous system
Schwann cell        Add myelin to axons in the peripheral nervous system

Gial cells perform functions such as:

Surround neurons and hold them in place
Supply nutrients and oxygen to neurons
Supply nutrients and oxygen to neurons
Destroy pathogens and remove dead neurons
Regulate the clearance of neurotransmitters from the synaptic cleft
Release gliotransmitters such as ATP, which modulate synaptic function.

Glial cells are known to be capable of mitosis whereas neurons usually are not.

In the brain "gray matter" is mostly neurons and "white matter" is mostly glial cells.

In the cerebral cortex the distribution of glia types is:

Oligodendrocytes   .756
Astrocytes         .173
Microglia          .065

                  Neurons   Glia
                  (109)     (109)
Cerebral cortex    16.3     60.8
Cerebellum         69.0     16.0


Nerve bundles


In a supercomputer the time to multiply two numbers is much shorter than the communication time with memory. Brains are the reverse. Signal speed is faster than computation. A neural signal travels 20 cm (all the way across the brain) during the time of one chemical synapse.

                CPUs   Flops   Devices         Cycles/second    Devices * Cycles/second

Brain            1      .1     1014 synapses       102                1016
Supercomputer   106    1016    106   CPUs          1010               1016
Flops = Floating point operations per second.

Human neurons are as small as physics will allow. If they were smaller then they would be close enough for signals to jump between them even without synapses.


Neurons do not undergo cell division. In most cases, neurons are generated by special types of stem cells. Astrocytes are star-shaped glial cells that have also been observed to turn into neurons by virtue of the stem cell characteristic pluripotency. In humans, neurogenesis largely ceases during adulthood; but in two brain areas, the hippocampus and olfactory bulb, there is strong evidence for generation of substantial numbers of new neurons.

Old-school arcade games

Old-school arcade games made the most out of the hardware, and they demanded precision to play.

Marble madness

Super Mario Brothers
Virtua Fighter
Hard Drivin'


Newton's laws and peripheral vision
Shotgun tracking
Team play

Realistic driving physics
Independent joysticks for motion and aiming
Agility required

Pong, the first commercial video game
Atari Pong console

Viola jokes

How do you keep your violin from getting stolen?
Put it in a viola case.

Why do violists leave their instrument cases on the dashboards of their cars?
So they can park in "handicapped" parking places. Or, if someone mistakes them for mafia, they might get some respect.

Why shouldn't violists take up mountaineering?
Because if they get lost it takes ages before anyone notices that they're missing.

What is the range of a Viola?
As far as you can throw it.

What's the difference between a chain saw and a viola?
If you absolutely had to, you could use a chain saw in a string quartet.

Did you hear about the violist who bragged that he could play 32nd notes? The rest of the orchestra didn't believe him so he proved it by playing one.

How many oboe players does it take to change a lightbulb?
Only one but he'll go through 100 bulbs to find the best one.

When Chuck Norris picked up a viola, violinists stopped telling viola jokes.

Table of contents

Main page

Support the free online science textbooks project

© Jason Maron, all rights reserved.