FIGURAS GEOMÉTRICAS CONSTRUCCION

Paper Models of PolyhedraGijs Korthals Altes

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

Copyrights © 1998-2001 Gijs.Korthals Altes All rights reserved . It's permitted to make copies for non-commercial purposes onlyemail: [email protected]

Paper Models of Polyhedra

Platonic Solids

Archimedean Solids

Kepler-Poinsot Polyhedra

Other Uniform Polyhedra

Compounds

Dodecahedron Cube and Tetrahedron Octahedron Icosahedron

Cuboctahedron Icosidodecahedron Truncated Tetrahedron Truncated Octahedron Truncated Cube Truncated Icosahedron (soccer ball) Truncated dodecahedron Rhombicuboctahedron Truncated Cuboctahedron Rhombicosidodecahedron Truncated Icosidodecahedron Snub Cube Snub Dodecahedron

Great Stellated DodecahedronSmall Stellated DodecahedronGreat Icosahedron Great Dodecahedron

TetrahemihexahedronOctahemioctahedronCubohemioctahedronSmall Rhombihexahedron

S mall DodecahemiododecahedronSmall Ditrigonal IcosidodecahedronGreat Dodecahedron

Stella OctangulaCompound of Cube and OctahedronCompound of Dodecahedron and IcosahedronCompound of Two CubesCompound of Three Cubes Compound of Five Cubes Compound of Five Octahedra Compound of Five Tetrahedra Compound of Truncated Icosahedron and Pentakisdodecahedron

Small Rhombidodecahedron

Other Polyhedra

Prism and Antiprism

Kaleidocycles

Other Paper Models

Pentagonal HexecontahedronPentagonalconsitetrahedronPyramidPentagonal PyramidDecahedronRhombic DodecahedronGreat RhombihexacronPentagonal DipyramidPentakisdodecahedronSmall TriakisoctahedronSmall Triambic IcosahedronPolyhedra Made of Isosceles TrianglesThird Stellation of the IcosahedronSixth Stellation of the IcosahedronSeventh Stellation of the IcosahedronEighth Stellation of the IcosahedronNinth Stellation of the IcosahedronFinal Stellation of the Icosahedron

Triangular PrismPentagonal PrismPe ntagonal AntiprismTriangular PrismOctagonal PrismOctagonal AntiprismPentagrammic PrismPentagrammic AntiprismHexagrammic PrismHexagrammic AntiprismTwisted Rectangular Prism

Hexagonal KaleidocycleOctagonal KaleidocycleDecagonal Kaleidocycle

Cylinder Tapered CylinderConeSpecial Cones"Matryoska house""Matryoska house" 50%GlobeChevaux-de-frise

12210

4

57

9.

6.

8

113

1 Dodecahedron

Dodecahedron

Dodecahedron

Cube

Tetrahedron

1

2

3 41 25 6

3

4 Cube

Tetrahedron

Octahedron

Icosahedron

Cuboctahedron

Icosidodecahedron

Truncated Tetrahedron

Truncated Octahedron

Truncated Cube

Truncated icosahedron

Truncated dodecahedron

Rhombicuboctahedron

Truncated cuboctahedron

Rombicosidodecahedron

Truncated Icosidodecahedron

Snub cube

Snub cubeRight-handed

Snub dodecahedron

Snub dodecahedronRight-handed

Small Stellated Dodecahedron

On this page a model outof one piece On the next pagesa model out of six pieces.Fold the long lines backwardsfold the short lines forwards

1

2

3

4

5

6

1

2

3 4

5

6

7

11

8

13

12

14

13

10

type 1

type 2

Octahemioctahedron

Cubohemioctahedron

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

A

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

B

C

DE

F

B

A

C

A

D

A

E

A

F

A

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

Small dodecahemiododecahedron

Great Stellated Dodecahedronmade out of one piece of paper.Cut the lines between the longand the short sides of the triangles.Fold the long lines backwards and fold the short lines forwards.

Great Stellated Dodecahedronmade out of two pieces of paper.Cut the lines between the longand the short sides of the triangles.Fold the long lines backwards and fold the short lines forwards.This is piece one. On the next pageis piece two.

1

1

Great Stellated Dodecahedronmade out of five pieces of paper.Cut the lines between the longand 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


N

P


N

P


N

P


N

P

Pentagonale hexacontahedron

1

2

3

4

5

6

7

8

9

10

12

Pentagonallconssitetrahedron

Stella OctangulaType 1Type 2

Fold lines of type 1backwardsFold lines of type 2 forwards

Fold the lineswith a rightangle backwards fold the other lines forwards

Compound of twoCubes

Compound of Three Cubes(Small version)Fold the dotted liines forwardsFold the other lines backwards

D

C

C

B

E

E

F

G

G

A

Compound of Three Cubes(Small version)Fold the dotted liines forwardsFold the other lines backwards

A

B

A

A

C

A

D

A

A

E

A

F

G

AA

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

G

G

BB

A

CC

D

D

EE

F

F

AA

B

C

D

A

A

AA

A

A

F

E

A

A

A

A

G

A H

M P

W a

Compound of five Octahedra

Color 1

If you use paper in five different colorseach octahedron has a different color

B

C

D

E

G

I

L

N O

F

Q

R

U

Y

d

c

J

K

V

X

T

Z

S

b

B K

N Q

U Z


Color 2

A

C

J

D

L

E

O

M

F

G

P

R

W

d

S

c

H

T

V

I

X

Y

a

b

C L

O R

V b


Color 3

A

B

K

DI

J

Q

G

H

S

M

E

F

N

P

Y

T

X

W

U

a

Z

d

c

D F

I S

X d


Color 4

A

E

B

G

H

C

J

K

L

O

P

U

R

T

a

Q

Z

V

W

Y

M

N

b

c

E G

J T

Y c


Color 5

A

D F

B

C

I

K

N

O

R

H

W

V

U

S

a

X

Z

M

L

b

d

P

Q

Pyramids

Pentagonal pyramid

Decahedron

Rhombic Dodecahedron

X

X

X

X

Great Rhombihexacron

Fold the short lines forwardsFold the long lines backwards

Pentagonal Dipyramid

X

X

X

X

Pentakisdodecahedron

‘Dodecahedron’

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

Hexakaidecahedron

‘Icosahedron’

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

Icositetrahedron

Icosioctahedron

Tricontidihedron

Tricontihexahedron

Tetracontahedron

Hecatohedron

Third Stallation of the Icosahedron

X

X

XX

Fold the dotted lines forwardsFold the other lines backwards

Third Stallation of the Icosahedron

X

X

XX


Sixth Stallation of the Icosahedron(small version)


First glue part AGlue the parts A-M on A

Part A:

XX

X

X

Parts B-M

F

B

C

D

E

A

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

A

B

Sixth Stallation of the Icosahedron(large version)

A

C


D

A


A

E


A F


Sixth Stallation of the Icosahedron(large)






A

B

C

Seventh Stellation of the Icosahedron

D

G

I

B

A

A

B

A

A

C

F

H

C

F

H

D

G

I

D

E

F

E

F

B

A

D

E

A

D

E

J

L

N

J

L

N

E

F

B


G

H

I

H

C

J

B

G

C

B

G

C

O

Q

R

O

Q

R

H

C

J


J

K

L

D

E

I

J

K

I

J

K

K

S

M

K

S

M

D

L

E

L


M

N

O

N

M

G

L

O

N

L

O

N

T

F

P

T

F

P

N

M

G


P

Q

R

Q

H

I

O

P

Q

O

P

Q

T

R

S

T

R

S

Q

H

I


S

T

K

R

M

R

M

T

S

T

S

K

P

P


B

C

C

D

D

E

E

FF

B

A

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

B

A

A


C

A

A


D

A

A


A

E

A


F

A

A


A

C

E

B

D

F

B

B

F

F

E

ED

DC

C

A

A

C

C

G

GK

KF

F

A

A

D

D

H

HG

GB

B

A

A

E

E

I

IH

HC

C

A

A

F

F

J

JI

ID

D

A

A

B

B

K

KJ

JE

E

Ninth Stellation of the IcosahedronFold the dotted lines forwardsFold the other lines backwards

K

I

G

L

J

HB

B

C

C

H

HL

LK

K

G

G

C

C

D

DI

IL

L

D

D

E

E

J

JL

H

H

K

K

L

L

I

IE

EF

F

B

B

F

F

J

JL

LG

G

G

G

H

H

I

IJ

JK

K

L

Final Stellation of the icosahedronFold the dotted lines forwardsFold the other lines backwards

AB

C

D

E

F

BA

F

K

G

C


CA

B

G

H

D


DA

C

H

I

E


EA

D

I

J

F


FA

E

J

K

B


GB

K

L

H

C


HC

G

L

I

D


ID

H

L

J

E


JE

I

L

K

F


KF

L

G

B

J


LG

J

I

H

K


Triangular prisms

Pentagonal Prism

Pentagonal Antiprism

Octagonal Prism

Octagonal Antiprism

Pentagrammic PrismFold the dotted lines forwardsFold the other lines backwards

Pentagrammic AntiprismFold the dotted lines forwardsFold the other lines backwards

Hexagrammic PrismFold the dotted lines forwardsFold the other lines backwards

Hexagramic AntiprismFold the dotted lines forwardsFold the other lines backwards

Twisted rectangular prism (45 degrees)

Twisted rectangular prism (90 degrees)

Twisted rectangular prism (+ 45 -45 degrees)

caleidocyclus

Caleidocyclus 8

Caleidocyclus 10

Cylinder

l = r 2 + h 2

c = 2 r

x = 360.2. . l / c

r1 = radiusr2 = radiusc = circumference of circle 1d = circumference of circle 2x =angel of the part of the large circlel = radius of the large circleh = height of the conei = heigt of the

tapered cylinder = pi = 3.1415

x

ln

r1

r2

1

d = c . n / l

Tapered Cylinder

x

l

r

l = r 2 + h 2

c = 2 r

x = 360.2. . l / c

r = radiusc = circumference of the circlex =angel of the part of the large circlel = radius of the large circleh = height of the cone = pi = 3.1415

Cone

Asymetric Cone

Square Cone

Square Cone

noordelijke, oostelijke muur huis en de bodem

north, east wall of the house and the floor

Paper Colour 1/ papier kleur1

zuidelijke en westelijke muur huis

dakkapel


south and west wall house

dormer-window

onderkantdak plus twee zijden

paper colour 2 / papier kleur 2

(uitsnijden / cut-out)

two sides of the roof

placedormer-window

Twee zijden dak

onderkant dak/


Two sides of the roof

bottom rooffits around walls don't glue roof on thewalls





dakkapel

dakkapel

dakkapel

dakkapel









dormer-window

dormer-window

dormer-window

dormer-window





























placedormer-window

placedormer-window

placedormer-window

placedormer-window

Twee zijden dak

Twee zijden dak

Twee zijden dak

Twee zijden dak

onderkant dak/

onderkant dak/

onderkant dak/

onderkant dak/













Globe

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1 1

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

2 2

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FIGURAS GEOMÉTRICAS CONSTRUCCION