STEM of Tai Chi: Mathematics

Jim Bruner | Mezzacello


Tai Chi is an art and it is a science. Mathematics is both as well. There is a great deal of math in Tai Chi. But like all wisdom it is hidden. You MUST understand the basics of ratios or fractions, the concept of patterns and algorithms, and the role of phi Φ in nature and in our own bodies. I am a math geek, but I am not a mathematician. I am an artist who loves math.
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Mathematics in Tai Chi

Tai Chi is full of beauty and math. It is easy to overlook the math because our culture is used to viewing pattern as a visual system. But pattern is  always a mathematical construct. It is pointless to argue this. the language of the universe is mathematics. Tai Chi embodies math through the patterns and structures of the 24 forms through space, time and symmetry. It is actually very beautiful and useful to think about Tai Chi as a spiritual, mathematical, and cultural construct simultaneously. It will actually make it easier to learn this way.

The role of Phi in the body, Tai Chi and the Universe

The golden spiral. a Rectangle that contains both a perfect square and perfect rectangle to infinity. Jim Bruner | Mezzacello

You may have never heard of phi (Φ) Phi (rhymes with Pie) is a ratio that is incredibly common in the universe. The approximate ratio of Φ is 1: 1.618 where the numbers repeat after the 8 into infinity. What most people don’t realize is that Phi is the universe. Most of us are familiar with the “Golden Rectangle” but do we really understand the ratio and proportion of this shape? It is reponsible for so much of the pattern in our universe.

  • The way electrons and protons in the atom interact is a variable of phi Φ
  • The way DNA in the cells of all life on earth spirals is a function of phi Φ
  • The seeds in a sunflower, aster, rose are expressed as phi Φ
  • The hair within a calix of hair and the way skin grows out from a baby’s head (The fontanelle) is phi Φ
  • The way a whirlpool forms, a hurricane spins, and a galaxy spirals. That’s all phi Φ
  • There is phi Φ in your body as well. It appears as the golden rectangle.

Phi in the human body

Find a pair of pants that fit you at the waist nicely. Fasten the button or clasp. Hold these pants up to your neck and wrap them around your neck. They should just touch. There is a relationship between your waist and your neck. There is also a relationship of proportion between your shoulders, legs, and feet. Leonardo DaVinci was highlighting this with his representation of the ideal human form in his Vitruvian Man. That’s all math. Let’s calculate your Golden Rectangle.

  1. Measure the width of your shoulder from the edge of the left to the edge of the right.
    • You can use any measure you want, English Standard inches or Metric CM. It does not matter.
    • Write down that number and it’s constituent units. This is your a reference.
  2. Multiply that number times phi Φ Shoulder width
    1. ‘S’ in this case is your shoulder width. My shoulder width is 44.5 cm (17.5″)
    2. The equation is: S*1.618= Golden Rectangle.
  3. Draw on the ground or tape that rectangle on the ground.
  4. Place both of your feet within the edge of one of the short edges of that rectangle.
  5. No step forward to the opposite corner of that rectangle, centering your torso in the middle of the rectangle.
  6. You will notice that your center of gravity is perfectly balanced.
    1. Explore this by moving your body around in a circle or crouching down.
  7. Now move one foot just a little within that rectangle or just a little outside.
    1. Now move your body. You should be very aware that you are out of balance.

Try to commit this feeling to memory. You will want to be within this golden rectangle whenever you do Tai Chi.

Parental note!

Teach Tai Chi to your kids who are experiencing rapid growth because of puberty or learning a new sport. They will benefit greatly from knowing what their natural balance when running, playing, flirting or just avoiding running into walls because their bodies are changing overnight.


There are four ratios in Tai Chi. Refresher: a ratio is a number that has a proportional relationship to another number. All fractions are ratios.  Phi Φ is a ratio (1:1.618) the fraction 1/4 can be written as 1:4 and it is still a ratio and a fraction. the four ratios for Tai Chi are:

  1. 100:1 (one foot flat one foot in the air)
  2. 50:50 (both feet flat on the ground with distance between them and the torso in perfect balance)
  3. 60:40 (one foot 3/4 (3:4) on the ground – but not flat – on the ground and the other just the toes and balls of the front of the foot – or just the heel)
  4. 80:20 ( one foot flat on the ground and just the toes or the heel of the foot touching)

All ratios equal 100 so they can be thought of as percentages as well. 80%:20% being an example. Why are ratios important in Tai Chi? the ratios relate to the amount of energy that is being placed on the entire foot, the toe or the ball of the human foot on the ground. When you were a kid you might have played the lava ground game, yes? ratios inform the body where the balance and pressure of the body should be in a particular moment. 80:20 allows the 20% foot to move quite quickly and the 80% foot to launch the body forward. 50%:50% is very stable. 100%:1 or 0 is the body in the air and NOT on the ground at all. 60%:40% is the workhorse of Tai Chi. Practice what this will feel like for you.

Algorithms and patterns

Netflix. We all of us know by this point what an algorithm is. The idea is that there is a relationship between the whole of a number and the pieces. In Tai Chi there are six algorithms:

  1. Hold the Moon
  2. Play the Lute (Guitar)
  3. Set the Trap
  4. White Crane Spreads its Wings
  5. Snake Creeps Down
  6. Apparent Close Up

What does it mean that these moves are algorithms? It means that all 24 moves in Yang Form Tai Chi are just modified components of these six forms. You will not believe me now, but it will become increasingly obvious. Some of the forms will become natural resting points as you learn the entire form – LEFT and RIGHT – because we live in a symmetrical universe – and they will be placeholders for you. That will be OK at first. But eventually you will have to perform Tai Chi as one fluid form. You will only need to know the algorithms at first. But they will be useful.

Fractals and repetition

This is a fractal. The shape and proportion repeats. It is also a representation of Phi. Jim Bruner | Mezzacello

Fractals are the way the blind universe assembles themselves. We will reference the transcendental numbers soon, but the universe and her twin nature do not waste energy – ever. If a pattern holds, it is repurposed. You may be aware of fractals already, but if you are not, it is good to familiarize yourself. In addition to being an algorithm, Tai Chi is also a fractal. It is composed of pieces and parts that repeat endlessly. You can for example perform all 24 of the Yang Form Tai Chi moves endlessly. They flow into each other. The fractal is balanced by what is called a “boundary set” usually 1, 2, 3 or 4 modes. Most of the fractals in the Yang Form are:

  1. (Commencing, Play the Lute*, Apparent Closeup*, Snake creeps out, searching for needle at the bottom of the sea, fan the back, block*, punch, parry, carry tiger to the mountain).
  2. (Grasping Sparrow’s Tail, Double Pulling, Snake creeps down, Golden Rooster, Fair Lady at the Shuttle, Block, Punch and Parry kind of qualify. All the double forms are symmetrical. In that there is a front/back or left/right symmetry. That is the algorithm and the fractal nature)
  3. (Parting Wild Horses Mane*, White Crane Spreads its Wings*)
  4. (Repulse Monkey and Wave Hands Like Clouds)

You may have noticed an asterisk (*) on some of these. These are algorithmic moves. They occur in modified form again and again.

Proportion and Exponents

We covered proportion in ratios above but it deserves another look. For example

  • Prime Numbers,
  • Boundary Sets, and
  • the Fibonacci Sequence.

Prime Numbers, Transcendental numbers, and Mystery

Pythagoras and the mystery numeric religions. Jim Bruner | Mezzacello

The transcendentals are bordering on the Metaphysics for me. Prime numbers are not transcendental, but they are magical. A prime number is any number that cannot be divided by any other number other than 1 or itself.  1, one, * is not a prime number. 1 and 0 are base pairs. There is only one even prime number. In all of infinity ∞ there is only ONE number that is even, and that is the number 2. 2 is also the base pair 1+2=3 in the natural order of things. followed by 3, 5 thse are the titans of the primes. ALL numbers within ∞ are disible by these numbers. these numbers are also the first 5 numbers in the Fibonacci sequence (covered below) 1+2=3, 2+3=5, n*∞… I think this is extraordinary. I come back to this when I need to think about something safe and concrete. Prime numbers never lie. Integers are incapable of deception.

So the actual Transcendentals

Phi Φ 1.618, Pi π 3.14 and the natural number e 2.718. So beautiful and mysterious. We do NOT know why they persist so tenaciously in our universe. Why for example is Phi automatically considered balanced and beautiful? Why does every circle in the universe have a diameter that is ALWAYS 3.14 times the length of its radius? Why does all exponential growth follow the natural number e? We cannot know. We must have faith.

  • Phi is related to truth and beauty and proportion.
  • Pi is related to the circle and the relationships of the squared numbers.
  • All exponential growth follows the ratio of e.

We do not know why. It just does. The greater mystery is that these numbers cannot be accurately portrayed as a number. Reason through this. We represent Phi as 1:1.618, Pi as 1:3.14 and e as 1:2.718. Bit NONE of these numbers are actual INTEGERS. And yet they work. The have a proportion to the ay we experience this 3D/4D reality and thus they matter when we want to be closer to the nature of peace and reality.

Math Matters.

Mathematical heroes

Amazing mathematicians have tried to elaborate on the mysteries:

  • Pythagoras – The pythagorean theorem a^2+b^2 = c^2
  • Archimedes – the rules of area and volume V = 4/3*πr^3
  • Plato – The platonic solids – Tetrahedron (4), Cube (6), Octahedron (8), Dodecahedron (12), Icosahedron (20)
  • Euclid – The Euclidean Geometry and the formulation of mathematical logic
  • Ilm al-jabr wa l-muqābala – Founder of “Al Gebr” or balanced equations that would become Algebra
  • Fibonacci – First mathematician to formalize e for exponential growth an = [Φn – (Φ)n] / Sqrt[5]
  • Fermat – Boundary sets in mathematics. This will become relevant in Metaphysics.
  • Liebniz – the creation of calculus and vector differentials d/dx ∫ Σ * Δ = F(r,t)
  • Newton – the creation of the differential equation to factor growth across time dy/dx = f(x)
  • Euler – The Euler identity and the introduction of the irrational number i e^√-1π + 1 = 0
  • Gauss – Magnetic Flux (How electricity propagates across a system) Φ = E*A*θ
  • von Neumann –  the origin for all quantum equations that I CANNOT represent in HTML! But they are gorgeous.
  • Turing – the founder of information theory as a mathematical modeling tool. The computer. You are welcome.
  • Penrose – The geometry of apace time and the relationship between π + Φ + θ and their relationship to Δ welcome to the fourth dimension of time…
  • Mandelbrot – Fractals and the shapes and patterns of the universe, Zn + 1 = Z(2/n) + c exponential growth and proportion in maths.
  • Witten – Fantastic mathematician responsible for the basis of super symmetry, string theory, M-theory and the emerging mathematics of consciousness.
  • Tegner – Holographic Theory the merging of all of these into a single theory of surface projection: dS = (ΔM  c₂)/T

These are superstars. Such amazing mathematicians. Mathematics is a language, a religion and secret that the universe holds close.

Symmetry and reality

By this point you know symmetry is a thing. You need to be comfortable with the fact that there are factor 2 Tai Chi moves that will have a left/right symmetry and a front/back symmetry. You will be able to see the instructor facing forwards, but when you perform a transformation (ie pivot 180 degrees) you will have to trust your instincts. You must RESIST the urge to look over your shoulder to see what the instructor is doing. You must KNOW that there are symmetrical versions of the same form.

In fact, just as in our universe there is an imaginary number line with a boundary set of [3] that looks like this:

∞, -3, -2, -1, 0, 1, 2, 3, ∞

So there is a left form Yang Tai Chi and a right form Yang Tai Chi. This may scare you. Do not let it. Once you know the form you will be able to see the algorithm of the way the body favors one side over another. For example in “Play the lute” in the right form, you will play the lute right-handed while the left hand holds the neck of the lute. In left form you will play the lute with the left hand while the right hand holds the neck of the lute. The same is true throughout. When you part wild horses mane in the right form, the right shoulder raises first, whereas in the left form the left shoulder rises first. This is an easy idea to imagine, but can be quite challenging in practice.
Next: Physics