Martial Arts Mathematics
I might as well admit it. I'm not good at math. I guess I'm more into poetry, writing, theater, and dance. So, it is even odd to me that I might write anything, on the Internet or even a scrap of paper, that has anything at all to do with math.
However, over many years studying martial arts, I have encountered certain recurring themes; some of them are rooted in mathematical principles and principles in science related to mathematics. Here, I would like to explore principles of mathematics that are found in martial arts and how it might be instructive for martial arts training.
Martial Arts Geometry
One of the first things a student of martial arts will notice in their training is the use of angles. The common angles used in martial arts liberally are the 90 degree angle and the 45 degree angle. These angles can be seen visually on the geometric shapes of triangles and squares. A right triangle has a 90 degree angle and a diagonal angle running along two points of the triangle.
In practical terms, the 90 degree angle, in relation to your position (in a fight, for instance), would be straight ahead of you, to your left and right, and directly in back of you. The 45 degree angles would be to the corners, two in front of you and two behind you; picture yourself in the center of a four-cornered room, facing a wall in the room: The four corners of the room are at 45 degree angles from where you are positioned.
Geometry: Degrees and Angles
When understanding movement, techniques, and positioning, these angles become very important.
Facing straight ahead, in this four-cornered room, movement to the left and right would be 90 degree turns; technically, to turn all the way around to face the back wall would be a 180 degree turn, which is two 90 degree turns. In common and simple usage, while training, speaking of a 90 (a direction which is 90 degrees from your position) is considered a direction straight in front, to the left or right, or in back of you.
However, when speaking of movement, one can speak of 45 degree turns (a movement towards the corner), 90 degree turns (movement to either side), 180 degree turns (turning around to face the opposite direction), or 360 degree turns (making a complete spin around to end up facing where you started). As might be expected, these movements can accompany footwork, kicks, striking, throws, and locks.
It should be noted that I've simplified these mathematical degrees to make it easier to understand in terms of martial arts training; it is simply easier to refer to the 90 degree angles as straight shots ahead, to the sides, and in back of you; while the 45 degree angles are simpler as references to directions towards the corners. Mathematically, however, these angles are based on the circle, which is measured as having 360 degrees: Straight ahead would be zero; straight to the right (if moving clock-wise) is the 90 degree angle; the right triangle is like a big wedge out of the circle, like you sliced a big hunk of pie; the 45 degree angle is halfway between zero and 90 degrees; half of 90 is 45, 45 two times is 90; as you continue the circle (from 90 degrees), another 45 degrees would be 135, which would be a back corner; go another 45 degrees and you have 180 degrees, which is the back wall, an about face; add another 45 degrees and you have 225 degrees and you'd face the other back corner (southwest). Add yet another 45 degrees and you are at 270 degrees, facing straight towards the left side from where you started; another 45 degrees and you have 315 degrees towards the front corner (from where you started), northwest; tack on another 45 degrees and you have 360 degrees, a full circle, facing where you started.
Related to footwork, the term "triangular footwork" is often used. In simple terms, this means moving in a 45 degree direction. If you picture yourself at the top of a triangle on the floor, the other two points of the triangle are points you can move to, at 45 degree angles. Movement in these directions could move you out of the way of an attack, at the same time opening a line of entry for you to attack your opponent. Behind you, likewise, are two other points at the bottom of another triangle, going at 45 degree angles, on which you can move to avoid attacks. Noticeably, this pattern forms a square, which is really made up of four triangles.
Along these lines, you can move yourself to a position in which you can have access to your opponent's targets. The other alternative is to move your opponent into a position in which his targets are available to you, and at which position you can use an angle to attack his targets which, when delivered, will have maximum results (do the most damage to your opponent to potentially end the fight). All of this involves your opponent's height, width and depth zones; you essentially want access to his dimensional zones while not giving him access to yours. The height zone is the vertical line from head to toe, the width zone runs side to side, and the depth zone goes in towards your opponent. The height zone can be heightened and lowered, the width zone can be moved side to side, and the depth zone can be penetrated. These zones can be manipulated in various ways, most often for the purpose of hindering your opponent's movement. Where his targets are most vulnerable is referred to as his center-line, which runs straight down the center of his body from head to toe; in practical terms, these are the targets on the front of his body and head: eyes, nose, philtrum nerve, cheek nerves, shins, abdomen, solar plexus, and groin; internal organs such as the heart, liver, spleen, and bladder; bones such as the ribs, collarbone, sternum, knee caps; and a variety of nerves all along the body; there are also targets on back of his body, also along his center-line, which are vital areas: The kidneys, the spine, the base of the skull, the tail-bone, are examples. Conversely, your own position should block off, in some way, all these targets from the opponent as much as is feasible. This is generally done by your own body being positioned mostly sideways in relation to your opponent, though slightly angled towards the 45, to allow you better mobility for using your own weapons; in addition, you should keep your back straight and head up.
Now, the arms and feet, while delivering strikes and kicks, also can travel on these angles. The more diversified the angles of your attacks, the better, because it gives greater access to targets on the opponent and potentially overwhelms him.
One of the main ideas behind the use of the triangle in martial arts is the principle of using a base to work from, in various ways. The most basic way this is used is in your stance. If you imagine your head is the top of the triangle and your two feet are the grounded two points at the bottom of the triangle, you will see that for you to keep your base, both points (your feet) must remain on the ground. Tip the triangle on one of its points, it no longer has a base. In simple terms, your balance is compromised when you're on one foot, so you don't want to stay in that position for long. Kicks can be used effectively, but it is not advised to stay on one foot for extended periods of time while fighting.
If you picture a paper cut out of a triangle sitting upright, facing you, you can see that it would be easy to knock it over with a push. However, if you turn the triangle sideways, you will see that if you push it, it will slide before it falls over.
Rear Naked Choke
Last point about triangles is the concept of three points of control, particularly used in grappling. It is common in martial arts to have three points of control to ensure that an opponent is controlled and a technique is pulled off effectively. Probably the best illustration of this is what they call the “Rear Naked Choke”. In this technique, one point of control is your arm on the opponent's neck, one of your hands on your own arm, and your other hand on your opponent's head. This ensures sinking that technique in and keeping control.
Jab-Cross-Hook Combination with Proper Stance and Body Mechanics
In addition, it should be noted that effective martial arts make use of both straight lines and circles. Even analyzing boxing techniques will reveal straight lines and circular movement being used. The standard jab-cross combination are two straight lines; the jab shoots straight out with the lead hand, then followed by a straight shot from the rear hand. Follow this up with the lead hand turning into a hook punch, then you have a circle added to the mix. In addition, footwork can also be along straight lines and circles, cutting to different angles.
The Physics of Martial Arts
Again, I'm certainly not a physicist, just like I'm not a mathematician. But understanding martial arts in terms of certain principles in physics can be instructive.
Physics makes use of mathematical formulas. One of the most relevant formulas of physics, in relation to martial arts, is the equation for momentum or kinetic energy; we can call it by different names, but in terms of martial arts, we are talking about power; power generated by your kicks, punches and strikes. The formula for kinetic energy is mass times velocity. Velocity is measured by distance and time: In other words, how much time it takes for something to go a certain distance. This is speed.
It is common in training to make sure as much of your body as possible and practical works in harmony with your strikes, punches, and kicks. Therefore, for instance, to add efficient power to a punch your whole body must move in line with your weapon. Also there are other factors that add power to a strike, such as gravity. Either way, your mass is moving harmoniously with the strike. As you might have guessed, speed adds to the power of the strike as well.
Along these lines, in physics there is the concept of vectors. A vector is the direction and magnitude of an object: Which direction it moves and how far. So, this is related to the principle discussed earlier about 90, 45, 180, and 360 degree angles: We can see that strikes and kicks can move towards these angles as well, and with harmonious use of mass along with speed, power can be generated behind these kicks and strikes.
Principles of Physics in Martial Arts
Forumula for Kinetic Energy
Power=Mass x Velocity
How much time it takes an object to go a certain distance
The direction and distance an object moves.
Kenpo: The Universal Pattern Uses Geometric Shapes to Illustrate Motion Along Different Planes
As well, it should be noted that geometry is involved in movement of all four limbs of the body: These parts, as discussed earlier, can move in straight lines, circles, and other shapes.
So, it seems demonstrable that concepts of mathematics, geometry, and physics are very helpful and instructive for learning martial arts skills. And, if you're like me, you might inadvertently learn about math while learning about martial arts.
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