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Complementary Solids
The purpose of this article is to introduce the topic I call Complementary Solids using the Platonic Solids as examples. The principle can be applied to any solid object whose surface consists entirely of corners, straight edges, and planes. 

Contents

  • Introduction to the Platonic Solids.
  • Some interesting relationships between these solids, and explaining the term Complementary Solids.
  • Euler's Equation.
  • Expanding on the theme of Platonic Solids.

The Platonic Solids

The five Platonic Solids, that is the regular solid figures whose faces are all the same regular plane figures, are:
  • The tetrahedron (four faces - all equilateral triangles).
  • The cube (six faces - all squares).
  • The octahedron (eight faces - all equilateral triangles).
  • The dodecahedron (twelve faces - all regular pentagons).
  • The icosahedron (twenty faces - all equilateral triangles).










Tetrahedron
 Cube
 Octahedron
 Dodecahedron
 Icosahedron

Defining Complementary Solids
The cube
Start with the cube, which is perhaps the most familiar of the Platonic solids. A cube has eight corners, twelve edges and six faces. Perform the following process on a cube.

The process

Stage 1
Take a cube and imagine filing away, equally, at all eight corners so that the newly-exposed faces are equilateral triangles. The original faces of the cube are now eight sided. Continue filing until these faces are regular octagons. We now have a figure whose faces are six regular octagons and eight equilateral triangles.

Stage 2
Continue filing the triangles until they just meet. The original faces of the cube are now diamonds (rotated squares). We now have a figure whose faces are six squares and eight equilateral triangles.

Stage 3
Continued filing changes the triangles into six sided figures. Continue filing until these faces are regular hexagons. We now have a figure whose faces are six squares and eight regular hexagons.

Stage 4
Continue filing until the squares disappear and the hexagons become equilateral triangles. We now have a regular octahedron.

Stage 5
Continued filing reduces the size of the octahedron but its shape does not change.

The result

A cube has eight corners, twelve edges and six faces. An octahedron has six corners, twelve edges and eight faces. The process has changed the corners into faces and the faces into corners.
Performing the process on the octahedron converts it back to a cube.

Definition

Let a pair of solid figures that exhibit this property be called Complementary Solids.

The cube and the octahedron
The cube and the octahedron are Complementary Solids.

Perform the filing process on the other Platonic Solids

The dodecahedron and the icosahedron
The dodecahedron and the icosahedron are Complementary Solids.

 The tetrahedron
The tetrahedron is complementary to itself.

Some properties of Complementary Platonic Solids

  • The faces, of one member of the complementary pair, are all either three, four or five-sided regular plane figures.
  • The faces, of the other member, are all equilateral triangles.
  • The number of four or five-sided regular plane figures meeting at a corner is always three (the number of sides of a triangle).
  • The number of triangles meeting at a corner is equal to the number of sides of a face of the complement.
  • Both members of the pair have the same number of even-dimensional elements, that is corners plus faces (Zero is classed as even).
  • Both members of the pair have the same number of odd-dimensional elements, that is edges plus spaces. (The surface of a three-dimensional figure separates an inner space from an outer space. So, there are two kinds of odd-dimensional elements associated with a three-dimensional figure: edges and spaces)

Euler's equation

The great 18th century mathematician, Leonard Euler, investigated solid figures, and devised an equation relating their corners, edges and faces. The edges must be straight.
  • Nc - Ne + Nf = 2
Where Nc is the number of corners, Ne is the number of edges and Nf is the number of faces.

I discovered this relationship before I heard about Euler's equation. I expressed it in a slightly different, and I think more elegant, way.
  • Nede = Node
Where Nede is the number of even-dimensional elements and Node is the number of odd-dimensional elements.

I wonder if the equation, put in this way, points to some deep truth about three-dimensional space.

My equation works for plane figures in three dimensional space, as well as for solids, whereas Euler's does not. Try it for yourself. Imagine a triangular plane figure suspended in three-dimensional space.

Why Euler's equation is relevant to complementary solids

We found that both members of a complementary pair have the same number of edges (odd-dimensional elements).

The process changes the corners into faces and the faces into corners (even dimensional elements). As the number of inner, and outer, spaces (the other odd-dimensional elements) does not change, Euler's Equation shows why the number of edges is the same for each member of the pair.

Why there are only five finite Platonic Solids

The faces of a Platonic Solid are identical regular plane figures. To form a corner, at least three faces must meet at a point. (Two faces meeting form an edge). The angles, at the corners of the faces, where they meet are equal because the faces are regular plane figures.

For the solid to be finite, these angles must total less than 360 degrees. So, an internal angle of a face must be less than 120 degrees. Note the case of a plane, tiled with hexagons whose internal angles are 120 degrees. Three hexagons fit together on the plane. If the angles total more than 360 degrees they will neither fit together on a plane nor at a corner.

The only regular plane figures whose internal angles are less than 120 degrees are the pentagon, the square and the equilateral triangle.
  • The internal angles of a regular pentagon are 108 degrees. 3 x 108 = 324, but 4 x 108 = 432. So, only three pentagons can form the corner of a solid figure.
  • The internal angles of a square are 90 degrees 3 x 90 = 270, but 4 x 90 = 360. So only three squares can form the corner of a solid figure.
  • The internal angles of an equilateral triangle are 60 degrees. 3 x 60 = 180, 4 x 60 = 240, 5 x 60 = 300, but 6 x 60 = 360. So, either three, four or five equilateral triangles can form the corner of a solid figure.
So, the only possible regular solid figures, whose surfaces consist of corners, straight edges and planes, have corners formed by either three pentagons, three squares, or three, four or five equilateral triangles. The Platonic Solids meet these criteria.

Limiting cases

The Platonic Solids are the regular solids whose faces all have either three, four or five sides.

What about a regular structure whose faces have six sides? This is the limiting case. It is infinite in size. Its surface is infinite in extent; an infinite plane, tiled with regular hexagons. I call this structure the Hexagon-tiled Megahedron.

The Hexagon-tiled Megahedron has a complement. It is also infinite in size and its surface is an infinite plane, tiled with equilateral triangles. I call this structure the Triangle-tiled Megahedron.

Just as three hexagons and six equilateral triangles fit together at a point on a plane surface, so do four squares. I call the structure, whose surface is an infinite plane, tiled with squares, the Square-tiled Megahedron.

Its complement consists of the same pattern of squares rotated by 45 degrees. So, the Square-tiled Megahedron is complementary to itself.

 It can be shown that, just like the finite structures, these complementary pairs have equal numbers of 'edges'.

Mike Holden - Nov 2005
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