Geometry is all around us, and we are surrounded by myriads of geometric forms, shapes, and patterns. Every living organism and all non-living things have an element of geometry within. Understanding the natural world requires an understanding of geometry.

Where there is a matter, there is geometry. 

For me, geometry is a fascinating subject. As the subject of learning, geometry requires the use of deductive reasoning, a logical process in which a conclusion is based on the multiple premises that are generally assumed to be true (facts, definitions, rules). Learning geometry also develops the visualization skills and spatial sense - an intuitive feel for form in space.

Geometry is essential in architecture and engineering fields, and mostly used in civil engineering. A thorough knowledge of descriptive geometry is definitely required for a civil engineer, and helps engineers to design and construct buildings, bridges, tunnels, dams, or highways. It has been, by the way, one of my favourite subjects during university years.

I use mathematics and geometry almost whole my life. To write about geometry in a way that does not look like a boring lecture is challenging. It is therefore my intention to write this post in a manner of picture book – geometry in words and pictures, because I prefer visualization wherever possible. I shall try to skip formulas, theorems, rules….. There will be only a little math.

PERFECT FORMS

Necessary Introduction

Polygons (many sides in Greek) are closed plane figures with straight sides. Regular polygons are polygons whose all sides are equal in length. A regular polyhedron (pl. polyhedra) is a three-dimensional solid which all faces are regular polygons.

Platonic Solids

These regular polyhedra are called the Platonic solids or perfect solids, named after the Greek philosopher Plato although he is not the first who described all of these forms. 

The Platonic solids are symmetrical geometric structures, which are bounded by regular polygons, all of the same size and shape. Moreover, all edges on each polygon are the same length and all angles are equal. The same number of faces meets at every vertex (corner or point).

Furthermore, if you draw a straight line between any two points (vertices) in any Platonic solid, this piece of straight line will be completely contained within the solid, which is the property of a convex polyhedron.

An amazing fact is that can be only five different regular convex polyhedra. These perfect forms are: 

Tetrahedron 

Octahedron 

Hexahedron (cube) 

Icosahedron 

Dodecahedron 

Platonic solids (Image source: www.joedubs.com)

Plato was deeply impressed by these forms and in one of his dialogues Timaeus he expounded a "theory of everything" based explicitly on these five solids. Plato concluded that they must be the fundamental building blocks—the atoms—of nature and he made a connection between five polyhedra and four essential (classical) elements of the universe;tetrahedron → fire, cube → earth, octahedron → air, icosahedron → water,and dodecahedron with its twelve pentagons was associated with the heavens and the twelve constellations.

Later, Aristotle, who had been Plato's student, introduced a new element to the system of the four classical elements. He classified aether as the “fifth element” (the quintessence). He postulated that the stars (cosmos itself) must be made of the heavenly substance, thus aether. Consequently, ether was assigned to the remaining solid --dodecahedron.

Why Only Five?

Geometric argument and deductive reasoning

Postulates: 

At least three faces must meet at each vertex to form a polyhedron. 

The sum of internal angles of polygons that meet at each vertex must be less than 360 degrees (at 360° they form the plane, i.e. the shape flattens out). 

If solid's faces that meet at each vertex are regular triangles, squares, and pentagons, the sum of angles at each corner is less than 360°. Forming regular solid of hexagons won’t work because hexagon has internal angles of 120°, and in the case where a minimum of 3 hexagonal faces meet at one vertex it gives → 3×120°=360°, thus the shape flattens out.

Consequently, there is no platonic solid formed from hexagons, or from any regular polygon of more than 5 sides.

The table below is a result based on previous arguments and reasoning. Solids are made only of regular triangles, squares, and pentagons. There are only five possibilities, thus five regular solids. Any other combination is not possible!

Mathematical proof given by Euler's formula confirmed that there are exactly five Platonic solids.

THE BEAUTY OF PLATONIC SOLIDS' GEOMETRY

Spheres and Platonic Solids

Each solid will fit perfectly inside of a sphere → circumsphere and all the angular points (vertices) are touching the edges of the sphere with no overlaps. Nonetheless, the inscribed sphere → insphere touches all the faces.

Inspheres of the Platonic solids (http://mathworld.wolfram.com)

Nested Platonic Solids

Platonic solids have the ability to nested one within the other. The corners of the inner Platonic solid touch the vertices or the edges of the outer solid. The amazing animation below shows the configuration of all five Platonic solids, each fits perfectly inside the other.

The video shows how (transparent) dodecahedron opens to reveal a cube inside, which opens to allow a tetrahedron to come out, then octahedron, which opens to reveal the inner icosahedron. All the Platonic solids are harmoniously nested one inside the other.

Duals of Platonic Solids

Each Platonic solid has a dual Platonic solid. If a midpoint (centre) of each face in the platonic solid is joined to the midpoint of each adjacent face, another platonic solid is created within the first.

It occurs in pairs between the solids when the number of faces in one solid = the number of vertices in another. 

The tetrahedron is self-dual (its dual is another tetrahedron), the only one with 4 faces and 4 points 

The cube and the octahedron form a dual pair (an octahedron can be formed from cube, and vice versa), 8 faces in cube=8 points in octahedron , or 6 points in cube = 6 faces in octahedron 

The dodecahedron and the icosahedron form a dual pair (a dodecahedron can be formed from an icosahedron, and vice versa), 12 faces in dodecahedron = 12 points in icosahedron , or 20 points in dodecahedron=20 faces in icosahedron 

In the image above is clearly visible how the octahedron occurs from the cube - putting a vertex at the midpoint of each face gives the vertices of dual polyhedron – octahedron. In vice versa, by connecting all midpoints of an octahedron’s faces occurs a cube, like in the image below.

Platonic solids duals (Image source: http://makerhome.blogspot.hr)

The Golden Ratio in Icosahedron and Dodecahedron

The icosahedron and his dual pair the dodecahedron are uniquely connected withthe golden ratio by virtue of three mutually perpendicular golden rectangles which fit into both. These mutually bisecting golden rectangles can be drawn connecting their vertices and mid-points respectively.