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The Maths of Aikido
The Maths of Aikido
by Niall Matthews
The Maths of Aikido

The only thing that makes me truly happy is mathematics. Snow, ice, and numbers.
Peter Høeg, Smilla's Sense of Snow

Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
Bertrand Russell, The Study of Mathematics

A root of an equation is a number which substituted into the equation instead of an unknown converts the equation into an identity. The root is said to satisfy the equation. Solving an equation implies finding all of its roots. An equation that is always satisfied, no matter the choice of values for its unknowns, is called an identity. (a + b)² = a²+ 2ab + b²
Stieg Larsson, The Girl Who Played With Fire

Charles, you are a mathematician. You're always looking for the elegant solution. Human behavior is rarely, if ever, elegant. The universe is full of these odd bumps and twists. You know, perhaps you need to make your equation less elegant, more complicated; less precise, more descriptive. It's not going to be as pretty, but it might work a little bit better.
Dr Larry Fleinhardt, Numb3rs

Everything is numbers
Charlie Eppes, Numb3rs
Maths has become cool in the last few years. And mathematicians have become stars. There was Will Hunting in Good Will Hunting. John Nash in A Beautiful Mind. Alan Turing in Breaking the Code. And the natural maths genius Lisbeth Salander in the Millenium trilogy by Stieg Larsson. Maths would have been a lot more fun at school if they had shown us episodes of Numb3rs.

We use geometry concepts everywhere in aikido. One point. Lines. Space. Planes. Curves. Vectors. Spheres. Spirals.

The triangle, the circle and the square are sometimes used as basic aikido symbols. They can have different meanings and there are a few suggestions in the links below. The idea of a square is not very useful in practical applications. But the triangle and the circle are very useful concepts.

There is an interesting judo textbook called The Secrets of Judo by Jiichi Watanabe and Lindy Avakian. It is unusual because it uses maths and physics to explain judo throws and holds. The parallelogram of forces, moment, the coefficient of friction…

There is more than a page of equations just about how to fall - ukemi. Time and velocity and acceleration and mass and distance. And the resisting force of the mat.

It's a good approach. As long as you also do the physical practice. Just look at the words. Physics. Physical practice.

It is not difficult to teach a judo throw by numbers. In uchikomi or nagekomi practice with a partner if you step in a certain way, move your body in a certain way and break the balance in a certain way finally you can do a technically accurate judo throw. It takes a long time to learn a throw well enough to be used against an opponent who is also trying to throw. But if you have explained it clearly a white belt can execute an effective throw.

Aikido is perhaps a little different. You can also explain an aikido technique and demonstrate it and break it down by numbers. But however clearly and well you explain an aikido throw to a white belt and however well the throw is done the result will still look like a throw by a white belt.

Jigoro Kano the founder of judo himself used a simple maths model to explain the principle of judo.

To understand what is meant by gentleness or giving way, let us say a man is standing before me whose strength is ten, and that my own strength is but seven. If he pushes me as hard as he can, I am sure to be pushed back or knocked down, even if I resist with all my might. This is opposing strength with strength. But if instead of opposing him I give way to the extent he has pushed, withdrawing my body and maintaining my balance, my opponent will lose his balance. Weakened by his awkward position, he will be unable to use all his strength. It will have fallen to three. Because I retain my balance, my strength remains at seven. Now I am stronger than my opponent and can defeat him by using only half my strength, keeping the other half available for some other purpose. Even if you are stronger than your opponent, it is better first to give way. By doing so you conserve energy while exhausting your opponent.

Jigoro Kano, Kodokan Judo
In aikido the maths are a little different. The aikido version goes like this. By me.

To understand what is meant by aikido let us say an attacker is standing before me whose physical strength is ten, and that my own physical strength is seven. If the attacker pushes hard I am sure to be pushed back or knocked down, even if I resist with all my might. This is opposing strength with strength. But if instead of opposing him I enter - or turn - or enter and turn - while keeping my own balance the attacker's balance will be completely broken. The attacker will be helpless and unable to use any strength. It will have fallen to nothing. Zero. My physical strength remains at seven. Now I am stronger than the attacker and can defeat him by using only a tiny fraction of my strength. But I do not try to defeat the attacker. I show the attacker that it is pointless to attack.
In Japan maths has a positive image. I have mentioned before that accountants are called keirishi. The shi uses the same character as bushi, samurai. I wrote about the great movie The Twilight Samurai in it's hard being a samurai. The hero of that movie is an accountant. But he is also a skilled swordsman. Another excellent samurai movie about an accountant is Abacus and Sword. It's about a samurai who earns respect with his ability as an accountant. Japanese students learn to use the abacus - soroban - in school. Ranking in soroban is by kyu and dan grades. The same as in the martial arts.




Mathematics in the movies


Mathematics and Art









The Twilight Samurai | Tasogare Seibei


Abacus and Sword | Bushi No Kakeibo

photo: Grand Challenge Equations by Duncan Hull

my blog on aikiweb | my blog on wordpress

© niall matthews 2012
Niall Matthews lives with his family in Japan. He teaches aikibudo and community self-defence courses and has taught budo for twenty-five years. He was the senior deshi of Kinjo Asoh Sensei, 7 dan Aikikai. He was the exclusive uke of Sadateru Arikawa Sensei, 9 dan Aikikai, at the hombu dojo in Tokyo for thirteen years until Arikawa Sensei's death in 2003. He has trained in several other martial arts to complement his aikido training, including judo (he has 4 dan from the Kodokan in Tokyo), kenjutsu (for about ten years) and karate (for about three years). He originally went to Japan as a staff member of the EU almost thirty years ago. He received 5 dan from Arikawa Sensei in 1995. This 5 dan is the last aikido dan he will receive in his life. His dojo is called Aikibudo Kokkijuku 合気武道克輝塾. Arikawa Sensei personally gave him the character for ki in kokki. It is the same character as teru in Sadateru - not the normal spelling of kokki 克己. It means you make your life shining and clear yourself.
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Old 01-02-2013, 01:28 AM   #2
Mario Tobias
Join Date: Aug 2006
Posts: 261
Re: The Maths of Aikido

Some aikido principles can be explained by physics. As an example, ALL techniques can be classified into one of the 3 simple levers. The lever has 3 components: the resistance, the force and fulcrum. The resistance is always the hara/center/dantien while the location of the fulcrum and force depends on the technique. The objective therefore is how to unbalance the resistance using any technique. This principle applies to all joints on a micro to macro level: micro for example are the fingers, thumbs (eg aiki age, aiki sage) macro being shoulder, elbow, neck (eg kihon waza).

Another example is the concept of non-resistance. This can be explained by the simple physics concept of work. The physics definition of work = Force X displacement X cos (angle). This is different from a layman's concept of work. The application to aikido is the force = line of KI and displacement is where you want uke/uke's body part to move. You are doing 0 work if the angle between uke's KI (line of force/movement) and where he's going is 90 degrees since cos(90) = 0. By doing 0 work, you are therefore using 0/minimal energy and also no power. This principle also applies on a micro/macro level and in all techniques.

Kuzushi is basically maximizing the effect of gravity in uke by tilting him from a perfectly upright position. A few degrees difference from the 90 degree position has a lot of effect from gravity.

By applying these physics concepts, you are literally integrating the "universe" within you since everybody succumbs to the physical laws. Nobody is immune. Imho, in aikido uke always has 2 opponents therefore will always be at a disadvantage: nage and the natural laws. Nage is just the "facilitator" applying the laws. IMO, in aikido nage's contribution is just a very tiny part in a perfect technique since it is the natural laws that uke succumbs to and not nage. Understand the natural laws and you will start understanding what aikido is IMHO. Aikido to me is basically applied physics in martial arts :-).

Last edited by Mario Tobias : 01-02-2013 at 01:35 AM.
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Old 01-02-2013, 05:55 AM   #3
Join Date: Sep 2008
Posts: 841
United Kingdom
Re: The Maths of Aikido

A more recent math hero is Nate Silver ( http://en.wikipedia.org/wiki/Nate_Silver ) of FiveThirtyEight fame.

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