The Geometry of Hyperparallelism

One of the most common families of non-Euclidean geometry is hyperbolic geometry – a self-consistent geometry of “obtuse” curvature.

While “normal,” foundational Euclidean geometry was founded upon five fundamental principles (Euclid’s five axioms), it is the fifth and final principle – the so-called parallel postulate – which is used to differentiate between this and any non-Euclidean space.

This principle defines, simply enough, that two lines are parallel to each other when a third line drawn between them will create angles which add up to exactly 180 degrees on each side. When any sort of space with curvature is considered, however, this postulate no longer holds true (necessarily), and therefore, an entirely new form of geometry must be instated. Continue reading “The Geometry of Hyperparallelism”

Invisible Wonders of the Human Body

Most people would agree that the physical human body is a wondrous creation. It is a self-regulating system which has automated its most essential functions, it can repair itself when injured, and it can defend itself against viruses and illnesses which would attack its internal organisms. Yes, this amazing vehicle, which we mostly take for granted, is a miracle of creation in itself, but when we look further and discover there are many other facets to the human body beyond what we can see and touch, then this wonderment turns to awe. Continue reading “Invisible Wonders of the Human Body”

The Most Important Figures in the History of Astronomy

Pythagoras (582 – 507 B.C)

A famous pre-Socratic era Hellenic philosopher, Pythagoras of Samos, Ionia, is for the most part best-known through his work as registered by his followers, the Pythagoreans. His followers thought that everything in the cosmos, including abstract thoughts, could be measured and carried in mathematical measures. Maybe what Pythagoras is most celebrated for is his uncovering of the Pythagorean theorem. Basically, the theorem assisted early mathematicians including Euclid to constitute presumptions and assist encourage the maths. In the area of the ascending scientific discipline of pre-astronomy, Pythagoras made-up a route for afresh astrology that enclosed an ingredient of practical maths, which would demonstrate assistive in the futurity of astronomy. Continue reading “The Most Important Figures in the History of Astronomy”

The Five Platonic Solids

A Universe Ruled by Geometry

Though not necessarily discovered by the legendary Greek philosopher Plato (some accounts give credit for this to the great Pythagoras, but history is unclear), the five geometrical shapes which have been classified as “regular” are also known as “platonic solids,” due to his intriguing analysis of them.

 

The Only Five

 

Though the idea of the existence of regular solids has been known since the earliest days of mathematics, it is not clear when the discovery of each of them was made. At the time of Pythagoras (sixth century B.C.), it is thought that only three of them were known, with the other two being discovered around the time of Plato – but like so many other things during this period, it is difficult to be sure.

 

So what are the solids, and why are they important?

 

The platonic solids are the only convex, three-dimensional geometrical figures (though there may exist more in higher dimensions) to possess the unique qualities of maintaining faces, edges, and angles which are all congruent over the entire body of the figure.

The five figures are, in order of their number of faces: The tetrahedron (made of three triangles), the cube (or hexahedron, made up of six squares), the octahedron (made of eight triangles), the dodecahedron (made of twelve pentagons), and the icosahedron (made of twenty triangles).

Early searches were made for further solids, but it was Euclid who is thought to be the first to have postulated that these are the only such figures which can exist – a fact which was later proved by mathematical means.

Historical Views of the Solids

 

The existence of these unusual, yet perfectly symmetrical, geometrical figures, led to several different interpretations over the years regarding their meaning (in some of the many attempts, with varying degrees of success, to associate vague mathematical principles with the physical universe over the centuries).

Plato, notably, devised a clever association with these solids (which, in part, is why they were later named for him). He ingeniously took the five solids and associated them with the four “elements” of the natural world – Earth, Fire, Air, and Water.

 

Plato argued that each of the elements could be conceived of as being composed of the first four solids – the tetrahedron was fire, the cube was earth, the octahedron was air, & the dodecahedron was water. The last solid, the icosahedron, was applied to the “heavenly sphere” upon which rested all of the stars and planets.

 

When Aristotle (fourth century, B.C.) further looked at the idea of the four elements and added a fifth aspect, the aether, he very well could have used this to apply the fifth Platonic solid, but he did not do so, instead looking at the mater a bit more “scientifically” (the word used, of course, relative to the “science” of his contemporaries).

 

Even two millennia later, with the rise of “modern” science during the Renaissance period, the platonic solids were still being applied to the universe.

Johannes Kepler (16th century), in particular, found the idea of the regular figures useful in describing the motion of the heavenly bodies. He even created a complicated model of the solar system, where each of the solids was laid into one of the other (like geometrical nesting dolls), separated by spheres circumscribed into each of them (an image of this can be found below).

Kepler associated each of the spheres surrounding solids (giving a total of six, counting the one outside the icosahedron) with the orbits of one of the known planets in the solar system – Mercury, Venus, Earth, Mars, Jupiter, and Saturn.

 

Today it is fairly clear that the planets are not so kind as to follow such perfectly geometrical, spherical paths, but few can doubt that it was a clever association to make. And, of course, Kepler did not stick with this idea for long – his unique perspectives into the motion of heavenly bodies eventually led him to his revolutionary Laws of Planetary Motion.

Platonic Solid

The Platonic solids also called the regular masses or regular polyhedra, are convex polyhedra with equal faces composed of congruent convex regular polygons. There are precisely five before-mentioned solids (Steinhaus 1999, pp. 252-256): the cube, icosahedron, dodecahedron, octahedron, & tetrahedron, as was demonstrated by Euclid in the last statement of the Elements. The Platonic solids are seldom too called “cosmic figures” (Cromwell 1997), although this term is seldom used to refer collectively to both the Platonic solids & Kepler-Poinsot solids (Coxeter 1973).

The Platonic solids were known to the ancient Greeks, & were described by Plato in his Timaeus ca. 350 BC. In this work, Plato equated the tetrahedron with the “element” fire,the icosahedron with water, the cube with earth, the octahedron with air, & the dodecahedron with the stuff of which the constellations & heavens were made (Cromwell 1997). Predating Plato, the Neolithic people of Scotl& developed the five solids a thous& years earlier. The stone models are held in the Ashmolean Museum in Oxford (Atiyah & Sutcliffe 2003).

Schläfli (1852) showed that there are precisely six natural materials with Platonic claims (i.e., regular polytopes) in four dimensions, three in five sizes, & three in all critical dimensions. However, his work (which contained no illustrations) continued virtually unknown until it was partially reprinted in English by Cayley (Schläfli 1858, 1860). Other mathematicians before-mentioned as Stringham subsequently discovered similar results separately in 1880 & Schläfli’s work was printed posthumously in its entirety in 1901.

The Math of the Solids

To date, it has been well proven that the five known solids are only which can exist in three-dimensional space (this has been proven in multiple ways – Euclid did so geometrically, while modern mathematicians have done so topologically).

 

They have been analyzed in great topological detail and have been completely defined mathematically in far greater detail than most people care to examine genuinely.

 

While it has become clear that the platonic solids do not necessarily rest at the heart of the physical universe, either in the construction of matter or the orbits of heavenly bodies, it has become clear that they do indeed exist in nature.

 

Living organisms on a microscopic level have been seen to confirm, at times, to the unique shapes offered by the platonic solids. Also, the crystal structures formed by certain atomic bonds are well-known to conform to shapes which echo the platonic solids, thanks to the strength provided by the regular symmetry of the shapes.

 

To date, the platonic solids have served mathematicians and physical scientists alike in numerous applications, and continue to remain interesting to students of mathematics, science, and history (both modern and ancient). Their value, in other words, should not be understated.