The Solids named after Plato and Archimedes
All of these solids are mathematically
associated with √3
In Plato’s Timaeus he explains the world according
to geometric shapes.
Platonic Solids
There are five Platonic solids. They are named for
the ancient Greek philosopher Plato who hypothesized in his dialogue,
the Timaeus, that the classical elements were made of these regular
solids 
The Timaeus makes conjectures on the composition of the four
elements which some ancient Greeks thought constituted the physical
universe: earth, water, air, and fire. Timaeus links each of these
elements to a certain Platonic solid: the element of earth would be a
cube, of air an octahedron, of water an icosahedron, and of fire a
tetrahedron. Each of these perfect polyhedra would
be in turn composed of triangular faces, the 306090 and the
454590 triangles.
Thirteen Archimedean
Solids
Equilateral triangles, and therefore, the value of √3 figure prominently
in the Thirteen Archimedean Solids
Nine of the thirteen Archimedean solids have some equilateral triangular
faces:
truncated cube (1), truncated tetrahedron (2), truncated dodecahedron
(3),
cuboctahedron (8), icosidodecahedron (9), (small) rhombicuboctahedron
(10),
(small) rhombicosidodecahedron (11), snub cube (12) and snub
dodecahedron
An Archimedean (semiregular) solid is a convex polyhedron composed of
two or more regular polygons meeting in identical vertices.
Archimedean Solids Each of these perfect polyhedra would be in turn
composed of triangular faces the a 45°  45°  90° Δ Triangle and the
30°  60°  90° Δ Triangle. These shapes also make extensive use of the
equilateral triangle, 60°  60°  60° Δ Triangle.
The height of this equilateral triangle is one half times the length of
the side times √3. (
½
x 2 x √3 = √3 )
This triangle can be divided into two identical 30°  60°  90° Δ
Triangles. And in the length of the longer leg of this triangle is equal
to the length of the shorter leg times √3.
Consider below the equilateral Triangle ABC , a 60°  60°  60° Δ
Triangle.
And the scalene right Triangle ABD, a 30°  60°  90° Δ Triangle.
