# 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.

# Físicos dizem ter detectado o quinto elemento – a quintessência

Redação do Site Inovação Tecnológica – 27/11/2020O observatório Planck rastreou a radiação cósmica de fundo, que os cientistas acreditam ser o «eco» do Big Bang. [Imagem: ESA/Planck]

O retorno do éter

Uma dupla de físicos da Alemanha e do Japão acredita ter dado um passo importante para ressuscitar uma das teorias mais controversas da Física: o éter.

Até Einstein, o éter era a substância essencial a partir da qual todas as partículas e ondas eram medidas, e no qual elas se deslocavam. Mas a teoria da relatividade especial dispensou o éter. Como defender o éter significava contrapor-se à relatividade, o termo foi logo banido e criou-se muito preconceito em torno dele.

Não têm faltado tentativas de ressuscitá-lo, sendo que a versão mais moderna equivale ao chamado vácuo quântico, que descreve o «vazio» como uma sopa de partículas que surgem e desaparecem o tempo todo – para quase todos os efeitos, aceitar o vácuo quântico significa apenas rebatizar o éter.

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# The Physics of Death (and What Happens to Your Energy When You Die)

## The Energy In You

Even though it’s an inexorable part of life, for many people, death — or at least the thought of ceasing to exist forever — can be a scary thing. The disturbing things that happen to the body during decomposition — the process by which cells and tissues begin to break down post mortem — are bad enough.

# The Biocentric Universe Theory: Life Creates Time, Space, and the Cosmos Itself

Adapted from Biocentrism: How Life and Consciousness Are the Keys to Understanding the True Nature of the Universe, by Robert Lanza with Bob Berman, published by BenBella Books in May 2009.

The farther we peer into space, the more we realize that the nature of the universe cannot be understood fully by inspecting spiral galaxies or watching distant supernovas. It lies deeper. It involves our very selves.

This insight snapped into focus one day while one of us (Lanza) was walking through the woods. Looking up, he saw a huge golden orb web spider tethered to the overhead boughs. There the creature sat on a single thread, reaching out across its web to detect the vibrations of a trapped insect struggling to escape. The spider surveyed its universe, but everything beyond that gossamer pinwheel was incomprehensible. The human observer seemed as far-off to the spider as telescopic objects seem to us. Yet there was something kindred: We humans, too, lie at the heart of a great web of space and time whose threads are connected according to laws that dwell in our minds.

# Why the number 137 is one of the greatest mysteries in physics

Famous physicists like Richard Feynman think 137 holds the answers to the Universe.

• The fine structure constant has mystified scientists since the 1800s.
• The number 1/137 might hold the clues to the Grand Unified Theory.
• Relativity, electromagnetism and quantum mechanics are unified by the number.

Does the Universe around us have a fundamental structure that can be glimpsed through special numbers?

The brilliant physicist Richard Feynman (1918-1988) famously thought so, saying there is a number that all theoretical physicists of worth should «worry about». He called it «one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man».

# There’s gold in your brain — we now know where it came from

• A unique, tiny grain of stardust has provided a look at the early universe.
• Computer simulations point to a single neutron-star collision as a significant source of heavy metals.
• Gold is more than bling — it’s in our neurons.
If you’ve got a thing for gold, you’d better have some money. Not only is the precious metal beautiful, but the amount of it in the universe is finite. A new study concludes that a single neutron star merger some 300 parsecs away produced a significant amount of it. «This means that in each of us we would find an eyelash worth of these elements, mostly in the form of iodine, which is essential to life,» says one of the astronomers involved in the study, Imre Bartos at the University of Florida.

Gold in particular is pretty fascinating stuff — it can even be argued that our individual existences depend on it, as astronomer Michelle Thaller explains.

# The Physics Still Hiding in the Higgs Boson

No new particles have been found at the Large Hadron Collider since the Higgs boson in 2012, but physicists say there’s much we can still learn from the Higgs itself.

# Einstein’s ‘Impossible’ Experiment Finally Performed

«Nunca digan que nunca sería posible «… «incensario pesimismo «….

A Hubble Space Telescope picture shows what’s known as an Einstein ring, when one galaxy bends and magnifies the light from a more distant one, as predicted by Einstein’s general theory of relativity. Astronomers have now used Hubble to see light from one star being bent by another in an experiment the famed physicist himself thought would be impossible to perform.