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Have you heard the term “regular” used in geometry before? This Euclidean term refers to polygons that are equiangular, meaning all angles are equal. In addition, all of the sides have the same length. These can take the form of a convex or star shape. Let’s take a look at some examples.

Triangle.Equilateral.svg

Equilateral Triangle

Regular polygon 4

Square

By User:Tomruen (Own work) [Public domain], via Wikimedia Commons

Regular Pentagon

Star polygon 5-2

Star Polygon

Regular Polyhedra

According to Wikipedia, “a regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitivevertex-transitive and face-transitive“.

Here are the five platonic solids – notice that the faces of the solid comprise of the same regular polygons.

Tetrahedron
(four faces)
Cube or hexahedron
(six faces)
Octahedron
(eight faces)
Dodecahedron
(twelve faces)
Icosahedron
(twenty faces)
Tetrahedron.svg by User:DTR
(Animation)
Hexahedron.svg
(Animation)
Octahedron.svg
(Animation)
POV-Ray-Dodecahedron.svg
(Animation)
Icosahedron.svg
(Animation) 

This Paper Models of Polyhedra site may interest you – the site allows you to download paper models for these solids and many more.

Regular Star Polyhedra

According to Wikipedia, “a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures“.

Small stellated dodecahedron.png Great dodecahedron.png Great stellated dodecahedron.png Great icosahedron.png
Small stellated dodecahedron
{5/2, 5}
Great dodecahedron
{5, 5/2}
Great stellated dodecahedron
{5/2, 3}
Great icosahedron
{3, 5/2}

Let’s take a look at some of these polyhedra in art and nature. Warning: not all of these examples are “regular”.

HPIM2402

Origami by Keri Valentine

BluePlatonicDice

Blue Platonic Dice

Football Pallo valmiina-cropped

Athens, GA by Keri Valentine

Athens, GA by Keri Valentine

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