The Calculating Fighter

Mathematics and the Martial Arts

You might ask yourself, ‘What relevance does mathematics have to the martial arts?’  Oddly enough, the answer is a great deal! 

To begin with, Maths is an integral part of Physics and so is enormously pertinent to those of us who study the combative sciences.  Learning to generate maximum force behind the strikes we deliver – whether punches, kicks or open-handed blows – requires an understanding of mechanics, both those of our own bodies and that of our target.  The delivery of a reverse punch, or a cross, is made efficient through ‘double-moment physics’ or ‘push-pull mechanics’, or to use another common term, ‘action/reaction’. 

Likewise, a roundhouse kick or a lateral elbow strike is made strong not just by using the right muscles in sequence, but by the application of rotational momentum.  Consequently ‘breaking’ techniques are a fine example of the application of maths.  A small mass, say the fist, under acceleration powers through an apparently rigid object (a square section of wood most commonly), all of which can be understood, no matter how counter-intuitive the event is, through the expression of a number of mathematical variables.

“Give me somewhere to stand, and I will move the world”, Archimedes

In the grappling arts, leverage is the single most important principle, as it tells us not only how to apply force, but also where the weak points of an opponent’s stance are, and those of our own.  In the Japanese arts of Judo and Ju-jitsu, kozushi – or ‘off-balancing’ – is the preliminary stage to executing a throw, and although it is not commonly expressed in mathematical terms, it is the science of mathematics in application that is at work here.  In both the Filipino art of Kalis and in Indonesian Silat, it is geometry – of the triangle and the pyramid – that allows us to knock the opponent down, simply by stepping through the hypotenuse of the imaginary triangle formed by his feet and perpendicular line descending from his centre of gravity.  The geometry of the relationship between angles, distance and arcs allows us to intercept and deflect our opponents’ attacks, while creating efficient trajectories for our own techniques and counters, whether empty-hand, or weapons strokes and thrusts also.

In any art in which stance training is given significance, it is that same understanding of geometry that underpins the use of each stance, each having an optimum application in supporting a movement of the upper body and a defined axis of ideal balance.

Rhythm and Broken Rhythm

Rhythm is as much a mathematical concept as a musical one, and indeed rhythm and cadence are expressed in terms of ratios – ¾ time etc. 

In Filipino Martial Arts (FMA), a common structure for drills in Panantukan, or Boxing, is to group techniques into what are referred to as ‘3-beats’.  One example is, what in my school we refer to as, ‘Catch jab, Return’.  In this drill, the pair practising stand in what we call a ‘matched stance’ i.e. they have the same lead.  One person initiates by throwing a jab to the chin of the other, while shifting their weight forward within the stance to lend weight to the technique, while the defender shifts their weight back and uses their rear hand to ‘catch’ the jabbing hand.  The defender then returns a jab to the person who initiated, who in turn ‘catches’ the counter-jab and counters that with another in the same manner.  There is a very short pause to distinguish one instance of the drill’s cycle from the next; the drill is then repeated, with the initiating role changing over to the other person.

The drill has a number of purposes, but the 3-beat structure accomplishes two things in particular:

  • Firstly, the training partners accomplish a lot of practise repetitions in a short, concentrated time.  In contrast, in a karate class, we might pair off, have the partners adjust their fighting distance –  maai* –  and uke (the initiator or provider of the attack) would execute the punch or other technique, with tori blocking or otherwise neutralising the threat, whereupon the pair would need to re-adjust and begin again, all of which takes time due to the precision of the ‘set-up’ phase.
  • Secondly, changing role at the beginning of, and within each cycle teaches a fluidity of mind-set.  If you are the initiator, you are throwing a jab, fully aware that your training partner (‘opponent’) will neutralise this and attempt a counter, at which point you must be ready to ‘counter-the-counter’.  Taking on the role of the person defending the first, exploratory jab – or any other technique performed to this pattern – you are just waiting for your chance to counter, again fully cognizant of the likelihood of this in its turn being countered.

This constantly alternating psychological state teaches the exponent to condition the opponent by offering a stimulus repeatedly.  By changing the anticipated follow-up, one type of ‘broken rhythm’ is produced – a psychological one.  In Kalis Ilustrisimo Repeticion Orihinal (KIRO) we don’t utilise this common drill structure, but we do utilise a variation of broken rhythm, which we call ‘praktion’ (fraction).  This is the concept of hitting – or rather ‘cutting’ in our case – in between the beats of the established rhythm, most commonly on the half-beat, but it can be any fraction of the timing depending on one’s skill level!

There are many other examples of the significance of mathematics, whether expressed in the geometry or the physics of applied movements within the arts – enough to fill a very large volume devoted to the subject, but I hope this indicates that there is something of interest to explore further.

 *’Maai’ (for which the kanji script is 間合い) is a little more than just an ‘ideal combative distance’, though this does bring together several of the ideas it encompasses.  It a mutable range, dependent upon your relative speed and reach in comparison to your opponent, and incorporates angle, which alters distance, and the timing required for any given technique to be delivered while allowing for the escape from a counter also.

©John Mellon M.A., and The Martial Arts University, 2011.  Unauthorized use and/or duplication of this material without express and written permission from this blog’s author and/or owner is strictly prohibited.  Excerpts and links may be used, provided that full and clear credit is given to John Mellon M.A. and The Martial Arts University with appropriate and specific direction to the original content.


About themartialartsuniversity

In daily training in the martial arts for the last 53 years; teaching almost daily for 43 years. Hold teaching ranks in 27 martial arts, Master ranked in 4 arts, Headmaster of 3 systems, the Founder of 2 arts, and the co-Founder of 2 others. Has been the official Technical Advisor to Profs. Wally and Leon Jay of Small Circle Ju-jitsu since the 1990's. Assistant Headmaster of Small Circle Ju-jitsu and the co-founder and joint Headmaster of Small Circle Concepts with Prof. Leon Jay. Has three Bachelor degrees and a Masters Degree in History. Experienced writer - published or featured in all of the UK martial arts magazines and in Inside Kung Fu in US, as well as: Esquire; She; Elle; Woman's Journal; Health and Fitness; Zest; Marie Claire and The Independent (UK national daily newspaper). Has been featured on BBC and ITV news programmes as expert on weapons defence and usage in relation to law enforcement issues. Former CPO (Bodyguard), and still active as a Bodyguard trainer.
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