Meissner Tetrahedra

faces

1

2

3

4

Reuleaux wedges

1

2

3

4

5

6

Meissner wedges

1

2

3

4

5

6

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The two Meissner bodies are solids of constant width. Others are spheres and certain solids of revolution.

The Reuleaux tetrahedron

T

is the intersection of four balls of radius 1, each centered at a vertex of a regular tetrahedron

R

with side length 1. Each of the six curved edges of

T

is the intersection of two spheres; three edges meet at each vertex and three surround each face.

For a curved edge

E

, let

F

be the corresponding straight edge of

R

and let

A

and

B

be the faces of

R

that meet at

F

. The planes containing

A

and

B

cut a wedge

V

out of

T

with edges that are circular arcs

C

and

D

. The wedge

W

is formed by rotating

C

into

D

around

F

. Rounding

E

means to replace

V

with

W

.

The first kind of Meissner body is obtained by rounding the three edges at a vertex of

T

and the second by rounding the three edges around a face of

T

.