poliedros
Transcript of poliedros
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
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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
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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
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
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
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
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
Pentagonale hexacontahedron
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2
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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
Compound of five Octahedra
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
Compound of five Octahedra
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
Compound of five Octahedra
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
Compound of five Octahedra
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
Fold the dotted lines forwardsFold the other lines backwards
Sixth Stallation of the Icosahedron(small version)
Fold the dotted lines forwardsFold the other lines backwards
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
Sixth Stallation of the Icosahedron(large version)
D
A
Sixth Stallation of the Icosahedron(large version)
A
E
Sixth Stallation of the Icosahedron(large version)
A F
Sixth Stallation of the Icosahedron(large version)
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)
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
Seventh Stellation of the Icosahedron
G
H
I
H
C
J
B
G
C
B
G
C
O
Q
R
O
Q
R
H
C
J
Seventh Stellation of the Icosahedron
J
K
L
D
E
I
J
K
I
J
K
K
S
M
K
S
M
D
L
E
L
Seventh Stellation of the Icosahedron
M
N
O
N
M
G
L
O
N
L
O
N
T
F
P
T
F
P
N
M
G
Seventh Stellation of the Icosahedron
P
Q
R
Q
H
I
O
P
Q
O
P
Q
T
R
S
T
R
S
Q
H
I
Seventh Stellation of the Icosahedron
S
T
K
R
M
R
M
T
S
T
S
K
P
P
Seventh Stellation of the Icosahedron
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
Eighth Stellation of the Icosahedron(Large version)Lold dotted lines forwardsFold other lines backwards
C
A
A
Eighth Stellation of the Icosahedron(Large version)Lold dotted lines forwardsFold other lines backwards
D
A
A
Eighth Stellation of the Icosahedron(Large version)Lold dotted lines forwardsFold other lines backwards
A
E
A
Eighth Stellation of the Icosahedron(Large version)Lold dotted lines forwardsFold other lines backwards
F
A
A
Eighth Stellation of the Icosahedron(Large version)Lold dotted lines forwardsFold other lines backwards
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
Final Stellation of the icosahedronFold the dotted lines forwardsFold the other lines backwards
CA
B
G
H
D
Final Stellation of the icosahedronFold the dotted lines forwardsFold the other lines backwards
DA
C
H
I
E
Final Stellation of the icosahedronFold the dotted lines forwardsFold the other lines backwards
EA
D
I
J
F
Final Stellation of the icosahedronFold the dotted lines forwardsFold the other lines backwards
FA
E
J
K
B
Final Stellation of the icosahedronFold the dotted lines forwardsFold the other lines backwards
GB
K
L
H
C
Final Stellation of the icosahedronFold the dotted lines forwardsFold the other lines backwards
HC
G
L
I
D
Final Stellation of the icosahedronFold the dotted lines forwardsFold the other lines backwards
ID
H
L
J
E
Final Stellation of the icosahedronFold the dotted lines forwardsFold the other lines backwards
JE
I
L
K
F
Final Stellation of the icosahedronFold the dotted lines forwardsFold the other lines backwards
KF
L
G
B
J
Final Stellation of the icosahedronFold the dotted lines forwardsFold the other lines backwards
LG
J
I
H
K
Final Stellation of the icosahedronFold the dotted lines forwardsFold the other lines backwards
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)
Kaleidocyclus
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
Paper Colour 1/ papier kleur1
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/
paper colour 2 / papier kleur 2
Two sides of the roof
bottom rooffits around walls don't glue roof on thewalls
zuidelijke en westelijke muur huis
zuidelijke en westelijke muur huis
zuidelijke en westelijke muur huis
zuidelijke en westelijke muur huis
dakkapel
dakkapel
dakkapel
dakkapel
Paper Colour 1/ papier kleur1
Paper Colour 1/ papier kleur1
Paper Colour 1/ papier kleur1
Paper Colour 1/ papier kleur1
south and west wall house
south and west wall house
south and west wall house
south and west wall house
dormer-window
dormer-window
dormer-window
dormer-window
noordelijke, oostelijke muur huis en de bodem
noordelijke, oostelijke muur huis en de bodem
noordelijke, oostelijke muur huis en de bodem
noordelijke, oostelijke muur huis en de bodem
north, east wall of the house and the floor
north, east wall of the house and the floor
north, east wall of the house and the floor
north, east wall of the house and the floor
Paper Colour 1/ papier kleur1
Paper Colour 1/ papier kleur1
Paper Colour 1/ papier kleur1
Paper Colour 1/ papier kleur1
onderkantdak plus twee zijden
onderkantdak plus twee zijden
onderkantdak plus twee zijden
onderkantdak plus twee zijden
paper colour 2 / papier kleur 2
paper colour 2 / papier kleur 2
paper colour 2 / papier kleur 2
paper colour 2 / papier kleur 2
(uitsnijden / cut-out)
(uitsnijden / cut-out)
(uitsnijden / cut-out)
(uitsnijden / cut-out)
two sides of the roof
two sides of the roof
two sides of the roof
two sides of the roof
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/
paper colour 2 / papier kleur 2
paper colour 2 / papier kleur 2
paper colour 2 / papier kleur 2
paper colour 2 / papier kleur 2
Two sides of the roof
Two sides of the roof
Two sides of the roof
Two sides of the roof
bottom rooffits around walls don't glue roof on thewalls
bottom rooffits around walls don't glue roof on thewalls
bottom rooffits around walls don't glue roof on thewalls
bottom rooffits around walls don't glue roof on thewalls
Globe
11
2
3
23
11
2
3
23
11
2
3
23
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
2
2
2
2
2
3
3
3
3
3
1 1
3
Large Chevaux-de-friseThe other two parts are on the next two pages
2 2
1
2 2
3