Martial arts and mathematics

martial arts - karate

Many basic principles from mathematics, physics and biomechanics can be found in the different martial arts and of course also in karate. This is mostly about power and energy transfer during shocks, kicks and levers or about angle information e.g. when changing direction within a kata or the correct execution of a technique (correct flexion of the respective joints).

In the following I would like to discuss some interesting connections between body, movement and mathematics, and how you can optimize your training (especially your karate training) afterwards.

For this I have to go a little further to get a certain basic understanding.

Table of contents:

The golden ratio - 0,618 and 0,382

In short, the golden ratio (for more information about the golden ratio, see Wikipedia) is the ratio of a distance (or other size) to a whole, where the ratio of the whole to its larger part corresponds to the ratio of the larger to the smaller part. The larger part corresponds to 61.8% (0.618) and the smaller part to 38.2% (0.382).

The golden section of the human body

kraken.jpgIn the most diverse books on biomechanics, citing works by Fischer, Hochmuth and Bernstein, relative weight values of body parts are given. (for values see also: - Section Mass distribution in the body).

The following are approximate values relative to total body weight: head 7%, torso 43%, upper arm 3%, forearm 2%, hand 1%, thigh 12%, lower leg 5% and foot 2%.

Now mathematics and the golden ratio come into play! If you add the masses of head, torso, upper arm, forearm and hand 2 times each, you get a value of 62%. The same for the lower body (consisting of 2 thighs, 2 lower legs and 2 feet) results in 38%. Amazing, isn't it? One can say that the ratio of upper body to lower body lies exactly on the golden section, namely 61.8% to 38.2%!

You can spin the whole thing even further. Both thighs together make up about 24% of the total body weight, and thus 61.8% of the lower body in relation to the rest of the leg (lower leg + foot = 38.2% of the lower body).

If one takes the mass of the two lower legs and the feet together (approx. 14% of the body weight), this results in a distribution of 61.8% again to the lower legs at the feet with 38.2%. Of course, this does not apply to all humans except for the decimal places, but on the whole the golden section describes these conditions surprisingly precisely.

The golden angle

The golden angle (see Wikipedia for more information about the golden angle) is obtained by applying the golden section to the full angle, i.e. 360 degrees. Thus, one gets 222.5 degrees (section 68.2%; 360 degrees times 0.618) and 137.5 degrees (section 38.2%; 360 degrees times 0.382 or again 222.5 times 0.618) rounded to a half degree accuracy.

The golden ratio of 137.5 degrees (times 0.618) is again rounded 85 degrees. Especially the 137.5 and the 85 degrees find meaningful application in karate without us consciously perceiving it. More information about this can be found in the next sections...

Karate and the golden ratio

Now we have learned a lot about the golden ratio, but where is the reference to karate or other martial arts, one or the other will ask?! Here are some examples! ๐Ÿ™‚

1.) Golden section at karate stands

First of all, probably the most common stand (at least in Shotokan) you learn first, the Zenkutsu-Dachi. At that time I learned the definition: 1 times hip/shoulder width, 1.5 times stride length. In the Shotokan, where long stands are required, it could also be 1.618 times more to stay with the golden ratio. ๐Ÿ™‚

This again results in the golden angle of 137.5 degrees between the thigh and lower leg. Amazing how I think!

Further the lower leg should stand 90 degrees to the underground. Since many Karateka tend to push the knee further forward and thus increase the 90 degree angle and ruin the knee in the long run, it might be a good idea to go to 85 degrees instead of 90 (golden ratio 61.8% of 137.5 degrees). Especially as you push your knee a few more degrees forward without worrying about it when walking forward, so that gravity can pull you forward into the next position. Shown by Inoue Yoshimi at one of his seminars.

The rear foot in Zenkutsu-Dachi should point as far forward as possible, and if at all only slightly to the side. Maximum 45 degrees? According to a smaller part of the golden section of 85 degrees (84.96 times 0.382) it should be 32.5 degrees. ๐Ÿ™‚

2.) Golden ratio in arm techniques in karate

kraken.jpgAngle information also plays a huge role in karate when it comes to arm techniques. Here are essentially 2 angles decided, which had been drilled into me very early. On the one hand the 90 degrees angle and 180 degrees (straight arm).

Especially the 90 degree angle between forearm and upper arm, e.g. when swinging a Soto-Ude-Uke, is interesting. Some karate teachers also say that the fist of the hand must be closer to the head so that the angle between forearm and upper arm is reduced to 45 degrees. Again, the golden section is in place without being known.

Because if we take 85 degrees instead of 90 as in the Zenkutsu-Dachi, or even the golden average of 85 degrees (85 times 0.618), we have an angle of 52.5 degrees for the swing-out movement, which can be called the optimum. This also beautifully describes an ideal ratio of the distance to be covered of the fist and the time. By 52.5 degrees one would shorten the distance in comparison to the 85-90 degrees, but at the same time still keep the distance so far that the mass of the fist can be accelerated optimally.

The principle of 90 or 85 degrees between the upper and lower arm is based on the lifting effect in many arm techniques. Example: If you want to hold a weight with an outstretched arm, it is more difficult than if you bend the arm and thus bring the weight closer to your own body center of gravity. Therefore one could also believe the golden ratio by saying that with block techniques (Soto-Ude-Uke, Uchi-Ude-Uke, Shuto-Ude-Uke) one deviates from the 90 degrees rather in the direction of 85 degrees than in the opposite direction.

The same applies to the Manji-Uke of the back arm. If you lift this over 90 degrees, it is no longer possible to tighten the latissimus (widest back muscle / Musculus latissimus dorsi), if you lower the arm a few degrees (to about 85) the muscle is fully contractile!

Another example would be the Oi-Zuki or Gyaku-Zuki (a straight fist punch). A stretched arm at impact (180 degrees -> 2x90 between forearm and upper arm) would be suboptimal, because the musculature wouldn't have any buffer to prevent an impact of the elbow joint. Many trainers point out that even with the Oi-Zuki you should leave a small bend in the joint, but mathematically considering the golden section you could assume 85 degrees by 2. So you get 170 degrees instead of 180, which is quite exactly what is meant. ๐Ÿ™‚

Conclusion - Closing remarks

Before I now throw around myself with countless examples, I leave it at that for now. Besides a large selection of books on the golden ratio books on the golden ratio, there are also an almost endless number of websites dealing exactly with this topic.

I think it is no coincidence that the golden ratio can be found everywhere in nature, architecture, music and everywhere else, so it should also be used in karate or also meditation exercises.

You could (if you haven't found your stand and stride length yet) simply take a folding rule, measure your stride length, and multiply that by 1.618 to see if you feel comfortable with this stride length in Zenkutsu-Dachi. Or multiply the stride length by 0.618 and test if this length is quite by chance comfortable for your personal Neko-Ashi-Dachi... If you also include the Fibonacci numbers, you can do a lot with them and open up new perspectives on training, movement, power and energy transfer.

kraken l.jpg

Of course, these are all just ideas and incentives, and it is up to each individual to decide whether or not to pursue them meticulously. All in all, I think it's definitely interesting!

Author: Sascha from

Author: Sascha

Some words about myself. My name is Sascha - i'm a software developer, trader and martial artist from germany. Besides programming, trading and martial arts some of my interests and hobbies are cooking, fitness and hearing loud heavy metal music. :D

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