[English/Japanese]
Introduction to Computational Origami : The World of New Computational Geometry
This page is the suppor page of the following book:
-
Ryuhei Uehara.
Introduction to Computational Origami : The World of New Computational Geometry.
238 pages, Springer, August 2020.
The details can be found on
the page by Springer.
It has
the page for PDF version.
Files of Figures
In this book, there are many nets that are hard to understand without cut and fold.
Some of them are put here for your personal use.
- Fig. 1.2:
A solid folded from the Latin cross.
It is hard to imagine the shape without folding.
- Fig. 2.6:
Some examples of tilings in the Alhambra.
It is located in Granada, Andalusia, Spain.
They say that all of seventeen different types of tilings can be found there.
- Fig. 2.8:
(a) A net foldable both of a regular octahedron and a tetramonohedron [O'Rourke 2000?].
(b) A net foldable both of a regular tetrahedron and a box [Hirata 2004].
(c) A net foldable both of a regular icosahedron and a tetramonohedron [Uehara 2010].
As you get used to them, it is easy to imagne, but
at first it is difficult unless without actual folding them.
- Fig. 3.1:
The simplest net that can fold to two different boxes.
It is my personal opinion that ``simplest''.
- Fig. 3.5:
A representative common net of two boxes.
- Fig. 3.7:
A common net of two boxes that is also a tiling.
We can also fold to a tetramonohedron in this way!
- Fig. 3.8:
A common net of two boxes of size 1×1×(2(j+1)(k+1)+3) and 1×j×(4k+5)
- Fig. 3.12:
A simple net of all 7 orthogonal polyhedra.
- Fig. 3.16:
A common net of
a box of size 1×1×5,
a box of size 1×2×3, and
a fake-box of size 0×1×11 (a.k.a. doubly covered rectangle).
- Fig. 3.17:
All of nine polyominoes of area 30 that can fold to
two boxes of size 1×1×7 and1×3×3, and
one cube of size √5×√5×√5.
- Fig. 3.18 and Fig. 3.19:
Two polyominoes of area 30 that can fold to
a box of size 1×1×7,
a box of size 1×3×3, and
a cube of size √5×√5×√5
in four different way of folding.
- Fig. 3.27:
A common net of three boxes of size
7×8×56, 7×14×38, and 2×13×58.
- Fig. 3.29:
A common net of three boxes of size
7×8×14, 2×4×43, and 2×13×16.
- Fig. 4.1:
A common net of a cube and an almost regular octahedron
(by Toshihiro Shirakawa, 2010).
- Fig. 4.10:
An example of a common net of a cube and an almost regular tetrahedron.
- Fig. 7.10:
An example of power diagram.
On the original petal polygon in red lines,
circles are drawn with radius corresponding to the petals
centered at the vertices of the base convex polygon.
Blue lines based on the crossing points of these circles
form the power diagram.
The blue lines forms a tree, which it gives us the (partial) ordering of gluing.
- Fig. 9.1:
An example of regular rep-cube of degree 2, which was the first rep-cube we found.
- Fig. 9.2:
A rep-cube of degree 4 (it is also uniform),
and a rep-cube of degree 5.
- Fig. 9.3:
A rep-cube of degree 8 and a rep-cube of degree 9,
which is also uniform.
- Fig. 9.4:
A rep-cube of degree 25.
All of eleven nets are used in this pattern.
It is built by the author (by his hand ;-).
- Fig. 9.5:
A part of rep-cube of degree 36.
This is useful since you can combine 6 copies of them.
- Fig. 9.6:
A rep-cube of degree 64.
When you build a cube, the left one is lid, four copies of the central one
form four sides, and the right one makes the base.
- Fig. 9.7:
A rep-cube of degree 50. All of eleven nets are used in this pattern.
Half is made by computer, and the other half is made by the author by his hand.
- Fig. 9.8:
A construction of regular and uniform rep-cubes.
(1) and (2).
Though each pattern has a parity depending on blue/red, the construction is simple.
- Fig. 9.15:
Non-regular rep-cubes for k=2 and k=10.
My colleague, Dr. Irina Kostitsyna made them by her hand.
- Fig. 9.17:
Five piece pattern for the pythagorean triple (3,4,5).
Using this five pieces, you can fold to
two cubes of size 3×3×3 and 4×4×4,
or you can fold to a cube of size 5×5×5.
- Fig. 9.19:
A net of a cube of size c×c×c.
The single red part can fold to a cube of size a×a×a.
When you fold a cube of size b×b×b using the blue part,
you have three holes on it, but they can be filled by the three light
blue parts.
- Fig. 9.22:
Five piece pattern for the pythagorean triple (5,12,13).
Using this five pieces, you can fold to
two cubes of size
5×5×5 and 12×12×12,
or you can fold to a cube of size 13×13×13.
- Fig. 10.1:
An edge unfolding of J17 that can fold to a regular tetrahedron in 3
different ways. This is the unique pattern for the property.
- Fig. 10.2:
An edge unfolding of J84 that can fold to a regular tetrahedron in 2
different ways. This is one of five patterns for the property.
- Fig. 10.3:
An edge unfolding of J86, which tiles plane in p2-tiling manner.
You can fold a tetramonohedron by following colors.
- Fig. 12.6:
An edge unfolding of J89, which tiles plane in p2-tiling manner.
You can fold a tetramonohedron by following blue parts.
Errata
None, so far :-)
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Links
Last modified: Sat Apr 25 23:26:41 JST 2015
by Ryuhei Uehara (uehara@jaist.ac.jp)
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