Euclid's Construction of a Regular Dodecahedron (XIII.17)

step

1

2

3

4

5

construct on all cube sides

Let

P

and

Q

be the centers of the two adjacent cube faces

BEFC

and

ABCD

with common side

BC

, and let

N

,

H

,

O

, and

L

be the midpoints of

BE

,

BC

,

CF

, and

AD

.

R

,

S

, and

T

are points that satisfy the golden section conditions

PR:RN=PN:PR

,

PS:SO=PO:PS

, and

QT:TH=QH:QT

.

UR

and

VS

are perpendicular to

BEFC

,

TW

is perpendicular to

ABCD

, and

UR=VS=TW=PR=PS=PR=QT

.

UVCWB

is one face of a regular dodecahedron.

The other faces are constructed in the same way.