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Polyhedra
plaited with paper strips
Tetrahedron,
Cube,
Cube (special),
Cube (2|x) "wrappings",
Octahedron,
Dodecahedron,
Icosahedron,
Cuboctahedron,
Truncated Tetrahedron,
Truncated Cube,
Truncated Octahedron,
Rhombicuboctahedron,
great Rhombicuboctahedron,
Snub Cube,
Icosidodecahedron,
Truncated Icosahedron,
Snub Dodecahedron,
Rhombic Dodecahedron, Calendar 2025,
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,
Kepler star,
Triakisoctahedron,
Tetrakishexahedron,
8-Cubes Object,
2-Cubes Object,
stellated Icosahedron,
Compound of 2 Cubes.
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").
An inner quadrilateral and an outer quadrilateral can be brought onto each other "parallel" (↑↑) or "antiparallel" (↑↓).
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 edge of the first inner quadrilateral
i 1 lies upon the last edge of the last outer quadrilateral of the strip. ("
closedness").
3. A plaiting strip should be
"consecutive", that means containing no quadrilaterals, with 2 neighbour edges belonging to the border of the strip.
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.).
Platonic solids and their paper strips for plaiting:
a tetrahedron plaiting with only one paper strip (mixed plain/vertex) can be found
here Hexahedron (cube) platonic
solid
8 vertices
12 edges
6 sides
6 squares
(6 sq.) plain
3 strips
4 sq.
a plain plaiting of the octahedron with only one paper strip can be found
here 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.
representation as knot
pattern
download: 2950 x 1480 pix, 217 kb:
ivn.gif
or as pdf:
ivn.pdf
Pseudoglobe plain
1 strip
(divided into
5 parts)
20 x 2 tr.
pattern as pdf-file:
icoerde.pdf
more polyhedra for plaiting:
Cube (special),
Cube (2|x) "wrappings",
Cuboctahedron,
Truncated Tetrahedron,
Truncated Cube,
Truncated Octahedron,
Rhombicuboctahedron,
Snub Cube,
great Rhombicuboctahedron,
Icosidodecahedron,
Truncated Icosahedron,
Snub Dodecahedron,
Rhombic Dodecahedron, Calendar 2025,
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,
Kepler star,
Triakisoctahedron,
Tetrakishexahedron,
8-Cubes Object,
2-Cubes Object,
stellated Icosahedron,
Compound of 2 Cubes.
last update: 2024/November/23
