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Hidden philosophy of the Pythagorean theorem
Pythagoras may have believed that the entire cosmos was constructed out of right triangles.
- Ancient Greeks believed that fire, air, water, and earth were the four elements of the universe.
- Plato associated these four elements with 3D geometrical solids.
- Pythagoras may have believed that the right triangle formed the basis of all reality.
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.
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So much for rest in peace.
- Australian scientists found that bodies kept moving for 17 months after being pronounced dead.
- Researchers used photography capture technology in 30-minute intervals every day to capture the movement.
- This study could help better identify time of death.
We're learning more new things about death everyday. Much has been said and theorized about the great divide between life and the Great Beyond. While everyone and every culture has their own philosophies and unique ideas on the subject, we're beginning to learn a lot of new scientific facts about the deceased corporeal form.
An Australian scientist has found that human bodies move for more than a year after being pronounced dead. These findings could have implications for fields as diverse as pathology to criminology.
Dead bodies keep moving
Researcher Alyson Wilson studied and photographed the movements of corpses over a 17 month timeframe. She recently told Agence France Presse about the shocking details of her discovery.
Reportedly, she and her team focused a camera for 17 months at the Australian Facility for Taphonomic Experimental Research (AFTER), taking images of a corpse every 30 minutes during the day. For the entire 17 month duration, the corpse continually moved.
"What we found was that the arms were significantly moving, so that arms that started off down beside the body ended up out to the side of the body," Wilson said.
The researchers mostly expected some kind of movement during the very early stages of decomposition, but Wilson further explained that their continual movement completely surprised the team:
"We think the movements relate to the process of decomposition, as the body mummifies and the ligaments dry out."
During one of the studies, arms that had been next to the body eventually ended up akimbo on their side.
The team's subject was one of the bodies stored at the "body farm," which sits on the outskirts of Sydney. (Wilson took a flight every month to check in on the cadaver.)Her findings were recently published in the journal, Forensic Science International: Synergy.
Implications of the study
The researchers believe that understanding these after death movements and decomposition rate could help better estimate the time of death. Police for example could benefit from this as they'd be able to give a timeframe to missing persons and link that up with an unidentified corpse. According to the team:
"Understanding decomposition rates for a human donor in the Australian environment is important for police, forensic anthropologists, and pathologists for the estimation of PMI to assist with the identification of unknown victims, as well as the investigation of criminal activity."
While scientists haven't found any evidence of necromancy. . . the discovery remains a curious new understanding about what happens with the body after we die.
Metal-like materials have been discovered in a very strange place.
- Bristle worms are odd-looking, spiky, segmented worms with super-strong jaws.
- Researchers have discovered that the jaws contain metal.
- It appears that biological processes could one day be used to manufacture metals.
The bristle worm, also known as polychaetes, has been around for an estimated 500 million years. Scientists believe that the super-resilient species has survived five mass extinctions, and there are some 10,000 species of them.
Be glad if you haven't encountered a bristle worm. Getting stung by one is an extremely itchy affair, as people who own saltwater aquariums can tell you after they've accidentally touched a bristle worm that hitchhiked into a tank aboard a live rock.
Bristle worms are typically one to six inches long when found in a tank, but capable of growing up to 24 inches long. All polychaetes have a segmented body, with each segment possessing a pair of legs, or parapodia, with tiny bristles. ("Polychaeate" is Greek for "much hair.") The parapodia and its bristles can shoot outward to snag prey, which is then transferred to a bristle worm's eversible mouth.
The jaws of one bristle worm — Platynereis dumerilii — are super-tough, virtually unbreakable. It turns out, according to a new study from researchers at the Technical University of Vienna, this strength is due to metal atoms.
Metals, not minerals
Fireworm, a type of bristle wormCredit: prilfish / Flickr
This is pretty unusual. The study's senior author Christian Hellmich explains: "The materials that vertebrates are made of are well researched. Bones, for example, are very hierarchically structured: There are organic and mineral parts, tiny structures are combined to form larger structures, which in turn form even larger structures."
The bristle worm jaw, by contrast, replaces the minerals from which other creatures' bones are built with atoms of magnesium and zinc arranged in a super-strong structure. It's this structure that is key. "On its own," he says, "the fact that there are metal atoms in the bristle worm jaw does not explain its excellent material properties."
Just deformable enough
Credit: by-studio / Adobe Stock
What makes conventional metal so strong is not just its atoms but the interactions between the atoms and the ways in which they slide against each other. The sliding allows for a small amount of elastoplastic deformation when pressure is applied, endowing metals with just enough malleability not to break, crack, or shatter.
Co-author Florian Raible of Max Perutz Labs surmises, "The construction principle that has made bristle worm jaws so successful apparently originated about 500 million years ago."
Raible explains, "The metal ions are incorporated directly into the protein chains and then ensure that different protein chains are held together." This leads to the creation of three-dimensional shapes the bristle worm can pack together into a structure that's just malleable enough to withstand a significant amount of force.
"It is precisely this combination," says the study's lead author Luis Zelaya-Lainez, "of high strength and deformability that is normally characteristic of metals.
So the bristle worm jaw is both metal-like and yet not. As Zelaya-Lainez puts it, "Here we are dealing with a completely different material, but interestingly, the metal atoms still provide strength and deformability there, just like in a piece of metal."
Observing the creation of a metal-like material from biological processes is a bit of a surprise and may suggest new approaches to materials development. "Biology could serve as inspiration here," says Hellmich, "for completely new kinds of materials. Perhaps it is even possible to produce high-performance materials in a biological way — much more efficiently and environmentally friendly than we manage today."
Dealing with rudeness can nudge you toward cognitive errors.
- Anchoring is a common bias that makes people fixate on one piece of data.
- A study showed that those who experienced rudeness were more likely to anchor themselves to bad data.
- In some simulations with medical students, this effect led to higher mortality rates.
Cognitive biases are funny little things. Everyone has them, nobody likes to admit it, and they can range from minor to severe depending on the situation. Biases can be influenced by factors as subtle as our mood or various personality traits.
A new study soon to be published in the Journal of Applied Psychology suggests that experiencing rudeness can be added to the list. More disturbingly, the study's findings suggest that it is a strong enough effect to impact how medical professionals diagnose patients.
Life hack: don't be rude to your doctor
The team of researchers behind the project tested to see if participants could be influenced by the common anchoring bias, defined by the researchers as "the tendency to rely too heavily or fixate on one piece of information when making judgments and decisions." Most people have experienced it. One of its more common forms involves being given a particular value, say in negotiations on price, which then becomes the center of reasoning even when reason would suggest that number should be ignored.
It can also pop up in medicine. As co-author Dr. Trevor Foulk explains, "If you go into the doctor and say 'I think I'm having a heart attack,' that can become an anchor and the doctor may get fixated on that diagnosis, even if you're just having indigestion. If doctors don't move off anchors enough, they'll start treating the wrong thing."
Lots of things can make somebody more or less likely to anchor themselves to an idea. The authors of the study, who have several papers on the effects of rudeness, decided to see if that could also cause people to stumble into cognitive errors. Past research suggested that exposure to rudeness can limit people's perspective — perhaps anchoring them.
In the first version of the study, medical students were given a hypothetical patient to treat and access to information on their condition alongside an (incorrect) suggestion on what the condition was. This served as the anchor. In some versions of the tests, the students overheard two doctors arguing rudely before diagnosing the patient. Later variations switched the diagnosis test for business negotiations or workplace tasks while maintaining the exposure to rudeness.
Across all iterations of the test, those exposed to rudeness were more likely to anchor themselves to the initial, incorrect suggestion despite the availability of evidence against it. This was less significant for study participants who scored higher on a test of how wide of a perspective they tended to have. The disposition of these participants, who answered in the affirmative to questions like, "Before criticizing somebody, I try to imagine how I would feel if I were in his/her place," was able to effectively negate the narrowing effects of rudeness.
What this means for you and your healthcare
The effects of anchoring when a medical diagnosis is on the line can be substantial. Dr. Foulk explains that, in some simulations, exposure to rudeness can raise the mortality rate as doctors fixate on the wrong problems.
The authors of the study suggest that managers take a keener interest in ensuring civility in workplaces and giving employees the tools they need to avoid judgment errors after dealing with rudeness. These steps could help prevent anchoring.
Also, you might consider being nicer to people.