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Polyhedra
plaited with paper strips

H. B. Meyer

Tetrahedron, Cube, Cube (special), Octahedron, Dodecahedron, Icosahedron, Cuboctahedron, Truncated Octahedron, Rhombicuboctahedron, great Rhombicuboctahedron, Snub Cube, Icosidodecahedron, Truncated Icosahedron, Snub Dodecahedron, Rhombic Dodecahedron Calendar 2015, Disdyakis Dodecahedron, Deltiodal Icositetrahedron, Pentagonal Icositetrahedron, Rhombic Tricontahedron, Small Stellated Dodecahedron, Great Dodecahedron, Johnson Solid #17, Johnson Solid #26, Johnson Solid #85, Triangular Prism, Square Antiprism, Pentagonal Prism, Pentagonal Antiprism, Hexagonal Prism, Hexagonal Antiprism, Square Deltohedron, t4-truncated Rhombic Dodecahedron, t4-truncated Deltoidal Icositetrahedron, dk4A4, a 12-sided solid, an 18-sided solid, bisymmetric hendecahedron, Nonahedron (african rattle).

Geometric Solids (Polyhedra) can be constructed by plaiting folded paper strips without use of glue.
Every strip is a sequence of certain quadrilaterals, and every folding-line is an edge or a diagonal of a quadrilateral.
The plaiting considered here shall be ruled by a "Plaiting Principle":
1. Always exactly two congruent quadrilaterals lie one upon another.

2. The quadrilaterals of every strip alternatively are situated inside
    ("inner quadrilateral" i) and outside ("outer quadrilateral").

a detailed example a small exhibition plaiting-scheme analysis

Ways of plaiting:

1. Every outer quadrilateral of a strip covers one or two sides of the polyhedron ("plain plaiting").

2. The border of a strip always connects a certain inner point ("midpoint") of a polyhedron-side and the midpoint of a certain edge of this side. ("edge plaiting").

3. The border of a strip always connects a certain inner point ("midpoint") of a polyhedron-side and a vertex of this side. ("vertex plaiting").

Tetrahedron, cube, octahedron, and icosahedron, for instance, can be plaited in every way, mixed forms are also possible (see: Cuboctahedron).

Requirements for plaiting strips:

1. The plaiting strips of a polyhedron should be congruent to each other, if possible ("congruence property").

2. Every plaiting strip should generate a "closed sequence of sides" of the polyhedron surface. That means: when plaiting is finished, the first inner quadrilateral i 1 and the last outer quadrilateral of the strip have a certain polyhedron edge in common. ("closedness").

3. A plaiting strip should, if possible, contain no "vertex quadrilaterals" (except the first inner quadrilateral and the last outer one), that means: no quadrilaterals, with neighbour edges belonging to the border of the strip ("vertex freeness").

4. the plaiting strips should have the ("maximum property"), that means they should not arise from cutting bigger strips of the same polyhedron into smaller ones (see edge plaiting of the cube e. g.).

Of course, these ideal requirements cannot be fulfilled always (the strips for the plain plaiting of the icosahedron, are not congruent, f. i.).

Platonic solids and their paper strips for plaiting:                                           
instruction: pattern printout
Tetrahedron

platonic
Solid

4 vertices

6 edges

4 sides
4 equi-
lateral
triangles
(4 tr.)
plain

2 strips,
mirror-
images

2 x 2 tr.


pattern download: 1060 x 620 pix, 18 kb: tg.gif
or as pdf: tg.pdf
edge

4 strips

3 x 2/3 tr.


pattern download: 1350 x 780 pix, 98 kb: tt.gif
or as pdf: tt.pdf
vertex

3 strips

4 x 2/3 tr.



pattern download: 2593 x 1761 pix, 100 kb: tk.gif
or as pdf: tk.pdf
Hexahedron
(cube)

platonic
solid

8 vertices

12 edges

6 sides
6 squares
(6 sq.)
plain

3 strips

4 sq.
edge

6 strips

4 x 1/2 sq.
vertex

4 strips

3 x 1/2 sq.

pattern download: 2801 x 1745 pix, 114 kb: str4.gif
or as pdf: str4.pdf. Easy version: he4.gif or he4.pdf
Octahedron

platonic
solid

6 vertices

12 edges

8 sides
8 equi-
lateral
triangles
(8 tr.)
plain

2 strips

4 x 2 tr.


pattern download: 2100 x 1000 pix, 36 kb: og.gif
or as pdf: og.pdf

Pseudoglobe
[Carlos A. Furuti]
plain

2 strips

4 x 2 tr.


pattern download: 2800 x 1637 pix, 510 kb: oer.gif
or as pdf: oer.pdf
edge

6 strips

4 x 2/3 tr.



pattern download:

1500 x 1040 pix,
60 kb: ok.gif
or as pdf: ok.pdf
vertex

4 strips

6 x 2/3 tr.


pattern download: 1620 x 1160 pix, 54 kb: oe.gif
or as pdf: oe.pdf
Dodeca-
hedron


platonic
solid

20 vertices

30 edges

12 sides
12 regular
pentagons
(12 pe.)


plaiting instructions
vertex

6 strips

10 x 2/5 pe.
a detailed guide to the vertex-plaiting of the dodecahedron is also given here (in German language).




pattern download: 2340 x 1370 pix, 74 kb: de.gif
or as pdf: de.pdf
egde

12 strips

5 x 2/5 pe.



pattern download: 2655 x 1971 pix, 88 kb: dk2.gif
or as pdf: dk2.pdf
vertex

6 strips

10 x 2/5 pe.


pattern download: 3249 x 2750 pix, 233 kB: dv.gif

or as pdf: dv.pdf
mixed
edge/vertex

6 strips

8 x 1/2 pe.


pattern download: 2889 x 2032 pix, 125 kB: doh.gif
or as pdf: doh.pdf
nice spiral:

plaiting instruction:
1) i16↑↑a3
2) i4↑↑a21, i21↓↑a16
3) i22↑↑a9, i9↓↑a4,
    i17↑↑a8
4) i10↑↑a27, i28↑↑a15, ..
vertex

1 strip
(divided
into 5 parts)

60 x 2/5 pe.
(12 x 2/5 pe.
per part)


pattern download: 3213 x 2200 pix, 236 kb: sr5.gif
or as pdf: sr5.pdf
Icosahedron

platonic
solid

12 vertices

30 edges

20 sides
20 equi-
lateral
triangles
(20 tr.)
plain

1 strip
(divided into
3 parts)

20 x 2 tr.

plaiting instruction: i3↓↑a6, i6↑↑a9, i9↓↑a2, i2↑↑a5, i10↑↑a3, i7↓↑a10, i5↓↑a8, i4↑↑a7, i8↑↑a1, and i1↓↑a4.

pattern download: 2950 x 1480 pix, 217 kb: ivn.gif
or as pdf: ivn.pdf

Remark: the plain plaiting of the icosahedron can be done with one single strip with 10 inner and 10 outer quadrilaterals (rhombi); use the above plaiting instruction. A nice plaiting exercise.


Pseudoglobe
plain

1 strip
(divided into
5 parts)

20 x 2 tr.


pattern as pdf-file: icoerde.pdf


plaiting instructions
vertex

6 strips

10 x 2/3 tr.


pattern download: 2400 x 1560 pix, 98 kb: ic.gif
or as pdf: ic.pdf
edges

12 strips

5 x 2/3 tr.



pattern
download:

1800 x 1050 pix,
65 kB: ik6.gif
or as pdf: ik6.pdf

mixed
edge/vertex

6 strips

4 x 2/3 tr.
6 x 1/2 tr.
1 tr.


pattern download: 1305 x 2017 pix, 120 kB: st6.gif
or as pdf: st6.pdf
more polyhedra for plaiting:
Cube (special), Cuboctahedron, Truncated Octahedron, Rhombicuboctahedron, Snub Cube, great Rhombicuboctahedron, Icosidodecahedron, Truncated Icosahedron, Snub Dodecahedron, Rhombic Dodecahedron Calendar 2015, Disdyakis Dodecahedron, Deltiodal Icositetrahedron, Pentagonal Icositetrahedron, Rhombic Tricontahedron, Small Stellated Dodecahedron, Great Dodecahedron, Johnson Solid #17, Johnson Solid #26, Johnson Solid #85, Triangular Prism, Square Antiprism, Pentagonal Prism, Pentagonal Antiprism, Hexagonal Prism, Hexagonal Antiprism, Square Deltohedron, t4-truncated Rhombic Dodecahedron, t4-truncated Deltoidal Icositetrahedron, dk4A4, a 12-sided solid, an 18-sided solid, bisymmetric hendecahedron, Nonahedron (african rattle).

last update: 2014/November/07





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