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A regular octahedron is an octahedron whose $8$ faces are all congruent equilateral triangles.

It has:

$6$ vertices
$12$ edges
$8$ faces

The regular octahedron is an example of a deltahedron.

Also known as

It is commonplace for authors to refer to a regular octahedron as just an octahedron, glossing over the fact of its regularity.

Also see

  • Results about regular octahedra can be found here.

Historical Note

Euclid in The Elements refers to this object just as an octahedron.

In the words of Euclid:

An octahedron is a solid figure contained by eight equal and equilateral triangles.

(The Elements: Book $\text{XI}$: Definition $26$)

According to the Pythagorean tradition, the regular octahedron was the symbol for the element air.

Linguistic Note

The word octahedron derives from the Classical Greek:

octo (ὀκτώ), meaning eight
hedron (a form of ἕδρα), meaning base or seat.

The technically correct plural of octahedron is octahedra, but the word octahedrons can often be found.