Definition:Platonic Solid
Definition
A platonic solid is a convex polyhedron:
- $(1): \quad$ whose faces are congruent regular polygons
- $(2): \quad$ each of whose vertices is the common vertex of the same number of faces.
That is, a platonic solid is a convex regular polyhedron.
Regular Tetrahedron
A regular tetrahedron is a tetrahedron whose $4$ faces are all congruent equilateral triangles.
Cube
A cube is a hexahedron whose $6$ faces are all congruent squares.
Regular Octahedron
A regular octahedron is an octahedron whose $8$ faces are all congruent equilateral triangles.
Regular Dodecahedron
A regular dodecahedron is a dodecahedron whose $12$ faces are all congruent regular pentagons.
Regular Icosahedron
A regular icosahedron is an icosahedron whose $20$ faces are all congruent equilateral triangles.
Also known as
Some sources refer to a Platonic solid as a regular polyhedron.
However, the latter term technically also encompasses polyhedra that are not convex.
Also see
- Results about platonic solids can be found here.
Source of Name
This entry was named for Plato.
Historical Note
The platonic solids were all well-known to the ancient Greeks.
The compass and straightedge constructions of the Platonic solids forms the climax of Euclid's The Elements.
Some have suggested that to glorify the Platonic solids was its primary purpose.
Some sources suggest that the construction of the final two of these were the work of Theaetetus of Athens, while others suggest that Hippasus of Metapontum may have contributed.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Platonic solid
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.)
- 2008: Ian Stewart: Taming the Infinite ... (previous) ... (next): Chapter $2$: The Logic of Shape: Euclid
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Platonic solid