Hidden philosophy of the Pythagorean theorem

In Plato’s dialogue, the Timaeus, we are presented with the theory that the cosmos is constructed out of right triangles.

This proposal Timaeus makes after reminding his audience [49Bff] that earlier theories that posited “water” (proposed by Thales), or “air” (proposed by Anaximenes), or “fire” (proposed by Heraclitus) as the original stuff from which the whole cosmos was created ran into an objection: if our world is full of these divergent appearances, how could we identify any one of these candidates as the basic stuff? For if there is fire at the stove, liquid in my cup, breathable invisible air, and temples made of hard stone — and they are all basically only one fundamental stuff — how are we to decide among them which is most basic?

A cosmos of geometry

However, if the basic underlying unity out of which the cosmos is made turns out to be right triangles, then proposing this underlying structure — i.e., the structure of fire, earth, air, and water — might overcome that objection. Here is what Timaeus proposes:

“In the first place, then, it is of course obvious to anyone, that fire, earth, water, and air are bodies; and all bodies have volume. Volume, moreover, must be bounded by surface, and every surface that is rectilinear is composed of triangles. Now all triangles are derived from two [i.e., scalene and isosceles], each having one right angle and the other angles acute… This we assume as the first beginning of fire and the other bodies, following the account that combines likelihood with necessity…” [Plato. Timaeus 53Cff]

A little later in that dialogue, Timaeus proposes further that from the right triangles, scalene and isosceles, the elements are built — we might call them molecules. If we place on a flat surface equilateral triangles, equilateral rectangles (i.e., squares), equilateral pentagons, and so on, and then determine which combinations “fold-up,” Plato shows us the discovery of the five regular solids — sometimes called the Platonic solids.

Three, four, and five equilateral triangles will fold up, and so will three squares and three pentagons.

If the combination of figures around a point sum to four right angles or more, they will not fold up. For the time being, I will leave off the dodecahedron (or combination of three pentagons that makes the “whole” into which the elements fit) to focus on the four elements: tetrahedron (fire), octahedron (air), icosahedron (water), and hexahedron (earth).

Everything is a right triangle

Now, to elaborate on the argument [53C], I propose to show using diagrams how the right triangle is the fundamental geometrical figure.

All figures can be dissected into triangles. (This is known to contemporary mathematicians as tessellation, or tiling, with triangles.)

Inside every species of triangle — equilateral, isosceles, scalene — there are two right triangles. We can see this by dropping a perpendicular from the vertex to the opposite side.

Inside every right triangle — if you divide from the right angle — we discover two similar right triangles, ad infinitum. Triangles are similar when they are the same shape but different size.

And thus, we arrive at Timaeus’ proposal that the right triangle is the fundamental geometrical figure, in its two species, scalene and isosceles, that contain within themselves an endless dissection into similar right triangles.

Now, no one can propose that the cosmos is made out of right triangles without a proof — a compelling line of reasoning — to show that the right triangle is the fundamental geometrical figure. Timaeus comes from Locri, southern Italy, a region where Pythagoras emigrated and Empedocles and Alcmaon lived. The Pythagoreans are a likely source of inspiration in this passage but not the other two. What proof known at this time showed that it was the right triangle? Could it have been the Pythagorean theorem?

Pythagorean theorem goes beyond squares

We now know that there are more than 400 different proofs of the famous theorem. Does one of them show that the right triangle is the basic geometrical figure? Be sure, it could not be a² + b² = c² because this is algebra, and the Greeks did not have algebra! A more promising source — the proof by similar right triangles — is the proof preserved at VI.31.

Notice that there are no figures at all on the sides of the right triangle. (In the above figure, the right angle is at “A.”) What the diagram shows is that inside every right triangle are two similar right triangles, forever divided.

Today, the Pythagorean theorem is taught using squares.

But, the Pythagorean theorem has nothing to do with squares! Squares are only a special case. The theorem holds for all figures similar in shape and proportionately drawn.

So, why the emphasis on squares? Because in the ancient Greek world proportional-scaling was hard to produce exactly and hard to confirm, and the confirmation had to come empirically. But squares eliminate the question of proportional scaling.

Pythagoras and the philosophy of cosmology

We have an ancient report that upon his proof, Pythagoras made a great ritual sacrifice, perhaps one hundred oxen. What precisely was his discovery that merited such an enormous gesture?

Could this review help us to begin to understand the metaphysical meaning of the hypotenuse theorem — namely, that what was being celebrated was not merely the proof that the area of the square on the hypotenuse of a right triangle was equal to the sum of the areas of the squares on the other two sides, but moreover, was the proof that the fundamental figure out of which the whole cosmos was constructed was the right triangle?

Prof. Robert Hahn has broad interests in the history of ancient and modern astronomy and physics, ancient technologies, the contributions of ancient Egypt and monumental architecture to early Greek philosophy and cosmology, and ancient mathematics and geometry of Egypt and Greece. Every year, he gives “Ancient Legacies” traveling seminars to Greece, Turkey, and Egypt. His latest book is The Metaphysics of the Pythagorean Theorem.

LINK ORIGINAL: https://bigthink.com/thinking/hidden-philosophy-pythagorean-theorem/#Echobox=1637200606

«Regular geometry»

«Regular geometry, the geometry of Euclid, is concerned with shapes which are smooth, except perhaps for corners and lines, special lines which are singularities, but some shapes in nature are so complicated that they are equally complicated at the big scale and come closer and closer and they don’t become any less complicated.» — Benoit Mandelbrot

Explore the fractal geometry of the universe in the free Unified Science Course at ResonanceScience.org

Link Original: Photo: spiderweb on my aloe vera
(photographer unknown: comment for credit)

What Is So Special About The Number 1.61803?

PHI(φ) is an irrational, non-terminating number as PI(π), but its significance is far more than PI(π) ;

Π = 3.14159265359…(pi)

Φ = 1.61803398874…(phi)

The Golden Ratio (phi = φ) is often called The Most Beautiful Number In The Universe.

The reason φ is so extraordinary is because it can be visualized almost everywhere, starting from geometry to the human body itself!

The Renaissance Artists called this “The Divine Proportion” or “The Golden Ratio”.

PHI(φ) can be seen appearing in the following ways:

1. Fibonacci Series




0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…

This series was developed by an Italian mathematician known as Leonardo Fibonacci. Other than the fact that each term is the sum of its two preceding consecutive terms, it can also be seen that if we divide a term greater than 2 by a term preceding it that the ratio always tends to 1.618…!

And if we continue this division after the 13th term we will always get a fixed number = 1.618


89/55 = 1.618

144/89 = 1.618

233/144 = 1.618

377/233 = 1.618

610/377 = 1.618

987/610 = 1.618

So on…!!!

2. The Human Body







  • For instance, if you divide the length from your head to toe by the length from your bellybutton to toe you will find the answer tending to φ.
  • Now, divide the length from your shoulder to the tip of the index finger by the length from your elbow to the wrist (of the same arm) and you’ll get φ again..!!
  • Divide the length from the top of the head to the shoulder by the length from your top of the head to your chin, φ again!
  • Top of your head to belly button by the length between you head and shoulder…..BANG….φ again!!!
  • Distance between your bellybutton and the knee, by the distance between knee and the bottom of the foot….φ again!
  • Now divide the length of your face to the width of the face……BAAM…φ again!!
  • Width of your two upper teeth to that of its height, and you’ll get φ again!
  • Lips to eyebrow divided by the length of the nose, φ again!

3. Plants

  • A sunflower grows in opposing spirals, the ratio of its rotation’s diameter to the next is 1.618…..i.e. φ again!
  • The ratio between the margin of a leaf to its veins(some plants) also gives φ.

4. DNA Of Organisms







  • DNA of the cell appears as a double-stranded helix referred to as B-DNA. This form of DNA has a two groove in its spirals, with a ratio of φ in the proportion of the major groove to the minor groove.
  • A cross-sectional view from the top of the DNA double helix forms a decagon. A decagon is actually two pentagons, with one rotated by 36 degrees from the other, so each spiral of the double helix must trace out the shape of a pentagon. The ratio of the diagonal of a pentagon to its side is φ to 1.

5. The Solar System

  • The average of the mean orbital distances of each successive planet in relation to the one before, tends to φ.
  • The Kepler’s Triangle(the triangle formed by utilizing the moon and the earth) is formed by a Pythagorean relation, in which the three sides of the right-angled triangle formed are always of this order:

Hypotenuse = φ

Perpendicular = √φ

Height = 1

  • If the rings of Saturn are closely looked at we will see that there is a ring that is quite denser than the other rings. Miraculously this inner ring exhibits the same golden section proportion as the brighter outer ring i.e. φ
  • Venus and the Earth are linked in an unusual relationship involving φ. If Mercury represents the basic unit of orbital distance and period in the solar system:

we find:

√Period of Venus * φ = Distance of the Earth

√2.5490 * 1.6180339 = 1.5966 * 1.6180339

= 2.5833 million kilometers

6. Art And Architecture








The Golden Ratio was probably most utilized by artists and architects while building their masterpieces. The following 5 pieces of work are specifically mentioned in the list as the golden ratio has been extensively used while creating them!

  • The Great Pyramid of Giza
  • Notre Dame
  • The Vitruvian Man
  • The Last Supper
  • The Parthenon

7. Music




If we divide an octave by a perfect fifth, (13/20) = φ

If we divide a perfect fifth by an octave, (8/13) = φ

If we divide a perfect fourth by a major sixth, (6/10) = φ

And if we divide a major third by a perfect fifth, (5/8) = φ

Therefore we can see that φ is indeed a mystical number which can be visualized all around us.

And if we observe closely we can find its traces going back before humanity was even inhabiting earth, for example, the skin folds of extinct dinosaurs, rare ancient insect segmentation, and much beyond that.

Link Original:https://medium.com/@gautamnag279/what-is-so-special-about-the-number-1-61803-7e0bbc0e89e2