Poliedros

Paper Models of Polyhedra Gijs Korthals Altes

Polyhedra are beautiful 3-D geometrical figures that have fascinated philosophers, mathematicians and artists for millennia

Copyrights Š 1998-2001 Gijs.Korthals Altes All rights reserved .

It's permitted to make copies for non-commercial purposes only email: gijs@korthals.altes.net

Paper Models of Polyhedra Platonic Solids Dodecahedron Cube and Tetrahedron Octahedron Icosahedron Archimedean Solids Cuboctahedron Icosidodecahedron Truncated Tetrahedron Truncated Octahedron Truncated Cube Truncated Icosahedron (soccer ball) Truncated dodecahedron Rhombicuboctahedron Truncated Cuboctahedron Rhombicosidodecahedron Truncated Icosidodecahedron Snub Cube Snub Dodecahedron Kepler-Poinsot Polyhedra Great Stellated Dodecahedron Small Stellated Dodecahedron Great Icosahedron Great Dodecahedron Other Uniform Polyhedra Tetrahemihexahedron Octahemioctahedron Cubohemioctahedron Small Rhombihexahedron Small Rhombidodecahedron S mall Dodecahemiododecahedron Small Ditrigonal Icosidodecahedron Great Dodecahedron Compounds Stella Octangula Compound of Cube and Octahedron Compound of Dodecahedron and Icosahedron Compound of Two Cubes Compound of Three Cubes Compound of Five Cubes Compound of Five Octahedra Compound of Five Tetrahedra Compound of Truncated Icosahedron and Pentakisdodecahedron

Other Polyhedra Pentagonal Hexecontahedron Pentagonalconsitetrahedron Pyramid Pentagonal Pyramid Decahedron Rhombic Dodecahedron Great Rhombihexacron Pentagonal Dipyramid Pentakisdodecahedron Small Triakisoctahedron Small Triambic Icosahedron Polyhedra Made of Isosceles Triangles Third Stellation of the Icosahedron Sixth Stellation of the Icosahedron Seventh Stellation of the Icosahedron Eighth Stellation of the Icosahedron Ninth Stellation of the Icosahedron Final Stellation of the Icosahedron Prism and Antiprism Triangular Prism Pentagonal Prism Pe ntagonal Antiprism Triangular Prism Octagonal Prism Octagonal Antiprism Pentagrammic Prism Pentagrammic Antiprism Hexagrammic Prism Hexagrammic Antiprism Twisted Rectangular Prism Kaleidocycles Hexagonal Kaleidocycle Octagonal Kaleidocycle Decagonal Kaleidocycle Other Paper Models Cylinder Tapered Cylinder Cone Special Cones "Matryoska house" "Matryoska house" 50% Globe Chevaux-de-frise

Dodecahedron

1 3 7 9.

5 12

2

10 8

11

4 6.

Dodecahedron

Dodecahedron

Cube

Tetrahedron

Cube

4 1

5

2

6

3 2

1

4

3

Tetrahedron

Octahedron

Icosahedron

Cuboctahedron

Icosidodecahedron

Truncated Tetrahedron

Truncated Octahedron

Truncated Cube

Truncated icosahedron

Truncated dodecahedron

Rhombicuboctahedron

Truncated cuboctahedron

Rombicosidodecahedron

Truncated Icosidodecahedron

Snub cube

Snub cube Right-handed

Snub dodecahedron

Snub dodecahedron Right-handed

Small Stellated Dodecahedron

On this page a model out of one piece On the next pages a model out of six pieces.

Fold the long lines backwards fold the short lines forwards

1

2

3

4

5

6

8

Octahemioctahedron type 1 type 2

13

7

11

12

5

6

2

1

10

3

4

14

13

Cubohemioctahedron

Small Rhombidodecahedron (small version) Fold the dotted lines forwards Fold the other lines

Small Rhombidodecahedron (large version) Fold the dotted lines forwards Fold the other lines

A B

C

F

E

D

B

A

C

A

D

A

E

A

F

A

Small dodecahemiododecahedron

Folds the lines between the triangles forwards. Folds the other lines backwards.

Great Stellated Dodecahedron made out of one piece of paper.

Cut the lines between the long and the short sides of the triangles.

Fold the long lines backwards and fold the short lines forwards.

Great Stellated Dodecahedron made out of two pieces of paper.

Cut the lines between the long and the short sides of the triangles.

Fold the long lines backwards and fold the short lines forwards.

This is piece one. On the next page is piece two.

1

1

Great Stellated Dodecahedron made out of five pieces of paper.

Cut the lines between the long and the short sides of the triangles.

Fold the long lines backwards and fold the short lines forwards.

N= Next part P= Previous part

N

P

Great Stellated Dodecahedron made out of five pieces of paper.

Cut the lines between the long and the short sides of the triangles.

Fold the long lines backwards and fold the short lines forwards.

N= Next part P= Previous part

N

P

Great Stellated Dodecahedron made out of five pieces of paper.

Cut the lines between the long and the short sides of the triangles.

Fold the long lines backwards and fold the short lines forwards.

N= Next part P= Previous part

N

P

Great Stellated Dodecahedron made out of five pieces of paper.

Cut the lines between the long and the short sides of the triangles.

Fold the long lines backwards and fold the short lines forwards.

N= Next part P= Previous part

N

P

Great Stellated Dodecahedron made out of five pieces of paper.

Cut the lines between the long and the short sides of the triangles.

Fold the long lines backwards and fold the short lines forwards.

N= Next part P= Previous part

N

P

Pentagonale hexacontahedron

1

2

3

4

5

6

7

8

9

10

12

Pentagonallconssitetrahedron

Stella Octangula Type 1 Type 2

Fold lines of type 1 backwards Fold lines of type 2 forwards

Fold the lines with a rightangle backwards fold the other lines forwards

Compound of two Cubes

Compound of Three Cubes (Small version)

Fold the dotted liines forwards Fold the other lines backwards

Compound of Three Cubes (Small version)

Fold the dotted liines forwards Fold the other lines backwards

D

A C

C

E

E

B

F

G G

A

B

C

A

A

A

D

E

A

A

A

F

A A

G

On this page a compound of five cubes made of one piece of paper.

On the next pages a compound of five cubes made of 7 pieces of paper

Instructions: Cut and fold the piece(s) of paper.

Glue the part without tabs around it last.

This is the top part of piece F. This one opposites the center part of piece A. Below an example of a part

Fold forwards Fold backwards

A

B B G

C C

G

D D

F

F

E

E

B

A A

C

A

A

D

A A

E

A

A

F

A A

G

A A

Compound of five Octahedra If you use paper in five different colors each octahedron has a different color

Color 1

A

H

D

E

G

R C

B

I

U

M

P

d

N

F

O Q

Y

L

c

W

K

a S

J

Z

X

V

T

b

Compound of five Octahedra Color 2

B

K

F

G

J

W D

A

L

C

N

Q

O

E

S

R

c

d

M

P

U

T

Z X

V

b

H

I

Y

a

Compound of five Octahedra Color 3

C

L

I

J

D

K

M

B

Y

A

O

R

P

Q

S

F

G

N

E H

V

X

b

W

a

T

U

Z

d

c

Compound of five Octahedra Color 4

D

F

K

L

O

P

B

A

G

E

I

S

H

R

Q

U C

T

a

J

X

Z

d c

Y V

W

b

M

N

Compound of five Octahedra Color 5

E

G

N

O

R

H

D

A

F

B

J

T

C

V

U

I

K

W

a

S

Y

X

c

Z

b L

M

d

P

Q

Pyramids

Pentagonal pyramid

Decahedron

Rhombic Dodecahedron

Great Rhombihexacron

X

X

X

X

Fold the short lines forwards Fold the long lines backwards

Pentagonal Dipyramid

Pentakisdodecahedron X X

X

X

‘Dodecahedron’ A convex dodecahedron (not a platonic solid) constructed of 12 isosceles triangles

Hexakaidecahedron

‘Icosahedron’

A convex icosahedron (not a platonic solid) constructed of 20 isosceles triangles

Icositetrahedron

Icosioctahedron

Tricontidihedron

Tricontihexahedron

Tetracontahedron

Hecatohedron

Third Stallation of the Icosahedron Fold the dotted lines forwards Fold the other lines backwards

X X

X

X

Third Stallation of the Icosahedron Fold the dotted lines forwards Fold the other lines backwards

X X

X

X

Sixth Stallation of the Icosahedron (small version) Fold the dotted lines forwards Fold the other lines backwards First glue part A Glue the parts A-M on A

Part A:

X X

X

X

Parts B-M

Sixth Stallation of the Icosahedron (large version) First glue the parts A until F Glue the 12 other parts on the ABCDEF

C

B

D

E

A

F

Sixth Stallation of the Icosahedron (large version)

B

A

Sixth Stallation of the Icosahedron (large version)

C

A

Sixth Stallation of the Icosahedron (large version)

D

A

Sixth Stallation of the Icosahedron (large version)

E

A

Sixth Stallation of the Icosahedron (large version)

A

F

Sixth Stallation of the Icosahedron (large)

Sixth Stallation of the Icosahedron (large)

Sixth Stallation of the Icosahedron (large)

Sixth Stallation of the Icosahedron (large)

Sixth Stallation of the Icosahedron (large)

Sixth Stallation of the Icosahedron (large)

Seventh Stellation of the Icosahedron

A

D

B

B D C

B

C

G

A

A G F

C

F

I

A

A I H

H

Seventh Stellation of the Icosahedron

D

E

A

A E J

E

J

F

D

D F L

F

L

B

E

E B N

N

Seventh Stellation of the Icosahedron

G

H

B

B H O

H

O

C

G

G C Q

I

Q

J

C

C J R

R

Seventh Stellation of the Icosahedron

J

D

I

I D K

K

K

L

J

J L S

L

S

E

K

K E M

M

Seventh Stellation of the Icosahedron

M

N

L

L N T

N

T

M

O

O M F

O

F

G

N

N G P

P

Seventh Stellation of the Icosahedron

P

Q

O

O Q T

Q

T

H

P

P H R

R

R

I

Q

Q I S

S

Seventh Stellation of the Icosahedron

S

K

R

R K T

T

T

P

M

M P S

S

Eighth Stellation of the Icosahedron (Large version) Lold dotted lines forwards Fold other lines backwards B

B

C

C

A

D

D

E

E

F F

Eighth Stellation of the Icosahedron (Large version) Lold dotted lines forwards Fold other lines backwards

B

A

A

Eighth Stellation of the Icosahedron (Large version) Lold dotted lines forwards Fold other lines backwards

C

A

A

Eighth Stellation of the Icosahedron (Large version) Lold dotted lines forwards Fold other lines backwards

D

A

A

Eighth Stellation of the Icosahedron (Large version) Lold dotted lines forwards Fold other lines backwards

E

A

A

Eighth Stellation of the Icosahedron (Large version) Lold dotted lines forwards Fold other lines backwards

A

A

F

Ninth Stellation of the Icosahedron Fold the dotted lines forwards Fold the other lines backwards

C

F

B

A B

A

F

A

C

B

F

C

E

G E

G D

K D

K C

F

B

C

A

A A

C

A

D

D

E

D

E

H

I H

I G

H G

H B

C

D

E

A

E

A A

F F

A

B F

B

J

K J

K I

J I

J D

E

K

L

B

G B

G

C

G

C

H

C

C

H

D H

D L

I L

I K

L

H

F

D

K D

I

K

J

E

L

E

L

J

I J

I L

E L

E H

F

G

K

B

K

G B

L F

G

H F

H

J

I J

I L

J L

J G

K

Final Stellation of the icosahedron Fold the dotted lines forwards Fold the other lines backwards

A B C

F

D E

Final Stellation of the icosahedron Fold the dotted lines forwards Fold the other lines backwards

B A F

C

K G

Final Stellation of the icosahedron Fold the dotted lines forwards Fold the other lines backwards

C A B

D

G H

Final Stellation of the icosahedron Fold the dotted lines forwards Fold the other lines backwards

D A C

E

H I

Final Stellation of the icosahedron Fold the dotted lines forwards Fold the other lines backwards

E A D

F

I J

Final Stellation of the icosahedron Fold the dotted lines forwards Fold the other lines backwards

F A E

B

J K

Final Stellation of the icosahedron Fold the dotted lines forwards Fold the other lines backwards

G B K

C

L H

Final Stellation of the icosahedron Fold the dotted lines forwards Fold the other lines backwards

H C G

D

L I

Final Stellation of the icosahedron Fold the dotted lines forwards Fold the other lines backwards

I D H

E

L J

Final Stellation of the icosahedron Fold the dotted lines forwards Fold the other lines backwards

J E I

F

L K

Final Stellation of the icosahedron Fold the dotted lines forwards Fold the other lines backwards

K F J

B

L G

Final Stellation of the icosahedron Fold the dotted lines forwards Fold the other lines backwards

L G K

H

J I

Triangular prisms

Pentagonal Prism

Pentagonal Antiprism

Octagonal Prism

Octagonal Antiprism

Pentagrammic Prism Fold the dotted lines forwards Fold the other lines backwards

Pentagrammic Antiprism Fold the dotted lines forwards Fold the other lines backwards

Hexagrammic Prism Fold the dotted lines forwards Fold the other lines backwards

Hexagramic Antiprism Fold the dotted lines forwards Fold the other lines backwards

Twisted rectangular prism (45 degrees)

Twisted rectangular prism (90 degrees)

Twisted rectangular prism (+ 45 -45 degrees)

Kaleidocyclus

Caleidocyclus 8

Caleidocyclus 10

Cylinder

Tapered Cylinder

l n x

l=

c=2

x = 360.2.

r

2

+

h

2

r1

. l /c

d=c.n/l r1 r2

r1 = radius r2 = radius c = circumference of circle 1 d = circumference of circle 2 x =angel of the part of the large circle l = radius of the large circle h = height of the cone i = heigt of the tapered cylinder = pi = 3.1415

Cone

l x

l=

c=2

x = 360.2.

r

r

2

+

h

2

r

. l /c

r = radius c = circumference of the circle x =angel of the part of the large circle l = radius of the large circle h = height of the cone = pi = 3.1415

Asymetric Cone

Square Cone

Square Cone

Paper Colour 1/ papier kleur1

noordelijke, oostelijke muur huis en de bodem

north, east wall of the house and the floor

Paper Colour 1/ papier kleur1

dakkapel

dormer-window

zuidelijke en westelijke muur huis

south and west wall house

paper colour 2 / papier kleur 2

(uitsnijden / cut-out)

onderkantdak plus twee zijden

two sides of the roof

place dormerwindow

onderkant dak/

bottom roof fits around walls don't glue roof on the walls

Twee zijden dak

Two sides of the roof

paper colour 2 / papier kleur 2

Paper Colour 1/ papier kleur1

Paper Colour 1/ papier kleur1

dakkapel

dormer-window

zuidelijke en westelijke muur huis

south and west wall house

noordelijke, oostelijke muur huis en de bodem

north, east wall of the house and the floor

Paper Colour 1/ papier kleur1

Paper Colour 1/ papier kleur1

dakkapel

dormer-window

zuidelijke en westelijke muur huis

south and west wall house

noordelijke, oostelijke muur huis en de bodem

north, east wall of the house and the floor

Paper Colour 1/ papier kleur1

Paper Colour 1/ papier kleur1

dakkapel

dormer-window

zuidelijke en westelijke muur huis

south and west wall house

noordelijke, oostelijke muur huis en de bodem

north, east wall of the house and the floor

Paper Colour 1/ papier kleur1

Paper Colour 1/ papier kleur1

dakkapel

dormer-window

zuidelijke en westelijke muur huis

south and west wall house

noordelijke, oostelijke muur huis en de bodem

north, east wall of the house and the floor

paper colour 2 / papier kleur 2

onderkant dak/

bottom roof fits around walls don't glue roof on the walls (uitsnijden / cut-out)

onderkantdak plus twee zijden

two sides of the roof

place dormerwindow Twee zijden dak

Two sides of the roof

paper colour 2 / papier kleur 2

paper colour 2 / papier kleur 2

onderkant dak/

bottom roof fits around walls don't glue roof on the walls (uitsnijden / cut-out)

onderkantdak plus twee zijden

two sides of the roof

place dormerwindow Twee zijden dak

Two sides of the roof

paper colour 2 / papier kleur 2

paper colour 2 / papier kleur 2

onderkant dak/

bottom roof fits around walls don't glue roof on the walls (uitsnijden / cut-out)

onderkantdak plus twee zijden

two sides of the roof

place dormerwindow Twee zijden dak

Two sides of the roof

paper colour 2 / papier kleur 2

paper colour 2 / papier kleur 2

onderkant dak/

bottom roof fits around walls don't glue roof on the walls (uitsnijden / cut-out)

onderkantdak plus twee zijden

two sides of the roof

place dormerwindow Twee zijden dak

Two sides of the roof

paper colour 2 / papier kleur 2

Globe

3

1

1

3 1

1 2

2

3 2

3

1

1

3 1

2

3

2

1

1

2

2

3

1 2

3

2

1

1

1

3

2

3

1

3

3

2

3

2

2

3 2 3 1

1 2

3 2

Large Chevaux-de-frise The other two parts are on the next two pages

3

1

1

1

2

2

3

2

2