Question

Find the symmetries of the Platonic Solids: Cube, Tetrahedron, Octahedron, Icosahedron, and Dodecahedron.

I. You need to describe with words the planes of reflectional symmetry and identify how many there are in any of the categories found.

II You need to describe with words the axes(lines) of rotational symmetry. For each type of axis, determine how many there are and the order of rotation.

Answer 1

A plane of symmetry divides a three dimensional shape into two congruent halves that are mirror images of each other.

This means that if you cut a 3 - D object from any side or angle and it turns out to be congruent with the other, its called a plane of symmetry.

a) Dual polyhedra

Every polyhedron has a dual (or "polar") polyhedron with faces and vertices interchanged. The dual of every Platonic solid is another Platonic solid, so that we can arrange the five solids into dual pairs.

• The tetrahedron is self-dual (i.e. its dual is another tetrahedron).

• The cube and the octahedron form a dual pair.

• The dodecahedron and the icosahedron form a dual pair.

If a polyhedron has Schläfli symbol {p, q}, then its dual has the symbol {q, p}. Indeed, every combinatorial property of one Platonic solid can be interpreted as another combinatorial property of the dual.

One can construct the dual polyhedron by taking the vertices of the dual to be the centers of the faces of the original figure. Connecting the centers of adjacent faces in the original forms the edges of the dual and thereby interchanges the number of faces and vertices while maintaining the number of edges.

More generally, one can dualize a Platonic solid with respect to a sphere of radius d concentric with the solid. The radii (R, ?, r) of a solid and those of its dual (R*, ?*, r*) are related by

d 2 = R ? r = r ? R = ? ? ? .

{\displaystyle d^{2}=R^{\ast }r=r^{\ast }R=\rho ^{\ast }\rho .}

Dualizing with respect to the midsphere (d = ?) is often convenient because the midsphere has the same relationship to both polyhedra. Taking d² = Rr yields a dual solid with the same circumradius and inradius (i.e. R* = R and r* = r).         

Dual Compounds

b) Symmetry groups

In mathematics, the concept of symmetry is studied with the notion of a mathematical group. Every polyhedron has an associated symmetry group, which is the set of all transformations (Euclidean isometries) which leave the polyhedron invariant. The order of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the full symmetry group, which includes reflections, and the proper symmetry group, which includes only rotations.

The symmetry groups of the Platonic solids are a special class of three-dimensional point groups known as polyhedral groups. The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the action of the symmetry group, as are the edges and faces. One says the action of the symmetry group is transitive on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is regular if and only if it is vertex-uniform, edge-uniform, and face-uniform.

There are only three symmetry groups associated with the Platonic solids rather than five, since the symmetry group of any polyhedron coincides with that of its dual. This is easily seen by examining the construction of the dual polyhedron. Any symmetry of the original must be a symmetry of the dual and vice versa. The three polyhedral groups are:

• the tetrahedral group T,

• the octahedral group O (which is also the symmetry group of the cube), and

• the icosahedral group I (which is also the symmetry group of the dodecahedron).

The orders of the proper (rotation) groups are 12, 24, and 60 respectively – precisely twice the number of edges in the respective polyhedra. The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts. All Platonic solids except the tetrahedron are centrally symmetric, meaning they are preserved under reflection through the origin.

The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parenthesis (likewise for the number of symmetries). Wythoff's kaleidoscope construction is a method for constructing polyhedra directly from their symmetry groups. They are listed for reference Wythoff's symbol for each of the Platonic solids.

We consider the symmetry of some basic geometric solids (convex polyhedra).
Important amongst these are the 5 Platonic solids (the only possible regular solids* in 3D): Ø Tetrahedron Ø Cube « Octahedron (identical symmetry) Ø Dodecahedron « Icosahedron (identical symmetry)      The symbol « implies the “dual of”.
Only simple rotational symmetries are considered (roto-inversion axes are not shown).
These symmetries are best understood by taking actual models in hand and looking at these symmetries.
Certain semi-regular solids are also frequently encountered in the structure of materials (e.g. rhombic dodecahedron). Some of these can be obtained by the truncation (cutting the edges in a systematic manner) of the regular solids (e.g. Tetrakaidecahedron, cuboctahedron)