This document discusses symbolic computations in conformal geometric algebra (CGA) for modeling three-dimensional origami folds. CGA provides an algebraic framework for representing geometric objects and transformations. The document introduces CGA, shows examples of visualizing geometric objects and operations in CGA, and demonstrates how origami folds can be formalized and visualized using CGA operations. Each origami fold is represented by an element of CGA, allowing the geometric properties of folds to be investigated using symbolic computations.

1. Symbolic Computations in
Conformal Geometric Algebra for
Three Dimensional Origami Folds
Yoshihiro
Mizoguchi
and
Hiroyuki
Ochiai
(Kyushu
University,
JAPAN)
July
20,
2016
COA2016
@
Vienna
University
of
Technology
hJps://github.com/KyushuUniversityMathemaQcs/MathemaQcaCGA

2. InsQtute
of
MathemaQcs
for
Industry,
Kyushu
University,
JAPAN.
Math-­‐for-­‐Industry
(MI)
is
a
new
research
area
that
serves
to
create
advanced
technologies
in
response
to
industrial
needs
by
innovaQng
flexible
&
versaQle
methods
in
mathemaQcs,
staQsQcs
&
compuQng.
Mission:
To
establish
a
leading-­‐edge
research
hub
for
mathemaQcs
as
applied
to
the
real
world
&
sciences,
while
nurturing
research
leaders
&
innovators
to
serve
societal
needs.

3. InsQtute
of
MathemaQcs
for
Industry,
Kyushu
University,
JAPAN.

4. MoQvaQon
O0=FirstO;
O1=Ori[O0, Ori2[P3, P1], {1}, π];
O2=Ori[O1, Ori2[P4, P3], {3}, π];
O3=Ori[O2, Ori2[P1, P4], {5}, −π];
O4=Ori[O3, Ori2[P5, P1], {4, 10}, π];
O5 = Ori[O4, Ori2[P6, P5], {13}, 0];
O6 = Ori[O5, Ori2[P7, P5], {29}, 0];
O7=Ori[O6 , Ori3[p2line[P8, P5],
p2line[P9, P8], 2], {27}, π];
O8=Ori[O7, Ori3[p2line[P8, P5],
p2line[P10, P8], 2], {59}, π];
O9=Ori[O8, Ori2[P11, P1], {15}, π];
O10=Ori[O9, Ori1[P7, P6], {31}, π];
O11=Ori[O10, Ori2[P7, P6], {11}, −π];
Formal
Origami
Design
Sheet
Origami
Instruc5on
Sheet
To
provide
a
formal
specificaQon
for
origami
to
invesQgate
its
geometrical
properQes
using
formal
symbolic
computaQons.
Each
funcQon
‘Ori#’
corresponding
to
a
fold
operaQon,
is
represented
by
an
element
of
CGA.

5. Table
of
Contents
• Algebra
for
Geometry
• CGA
(Conformal
Geometric
Algebra)
– DefiniQons
– Examples
– VisualizaQon
– Equality
Check
(Algebra)
• Origami
Folds
– FormalizaQon
– VisualizaQon
– 3D
Folds
• Concolusion

6. Algebra
for
Geometry
• Complex
Numbers
(2-­‐dim)
• Quaternion
(3-­‐dim)
• Conformal
Geometric
Algebra
(n-­‐dim)

7. Algebra of geometric operators (1)
7

8. Algebra of geometric operators (2)
8

9. CGA (Conformal Geometric Algebra)
9

10. R[E] (Product)
A
MathemaQca
module
for
Conformal
Geometric
Algebra
and
Origami
Folding
10
eφ ∗ eS = eS ∗ eφ = eS
e{0} ∗ eS =
e{0}∪S 0 ∉ S( )
0 0 ∈ S( )
'
(
)
e{a} ∗ eS =
(−1)
s∈S |s<a{ }
e{a}∪S a ∈{1,2,3} ∧ a ∉ S( )
(−1)
s∈S |s<a{ }
eS −{a} a ∈{1,2,3} ∧ a ∈ S( )
'
(
,
),
e{∞} ∗ eS =
(−1) S
e{∞}∪S 0 ∉ S ∧∞ ∉ S( )
0 0 ∉ S ∧∞ ∈ S( )
−e{0} ∗ e{∞} ∗ eS −{0}( ) − 2eS −{0} 0 ∈ S( )
'
(
,
)
,
eT∪{a} ∗ eS = eT ∗(e{a} ∗ eS ) (∀t ∈T , t < a)

11. R[W] (Outer Product and Inner Product)
11

12. Example (1)
12

13. Example (2)
13

14. VisualizaQon
(CGA)

15. Objects in R3
15

16. Operators in R3
16
-50
0
50
100
-100
-50
0
50
100
-100 -50 0 50 100
-100
-50
0
50
100
translator
rotor
dilator

17. 17
X ∧P x, y, z( )= αS wS
S ⊂ 0,1, 2, 3, ∞{ }
∑ = 0 ⇔∀S ⊂ 0,1,2,3,∞{ },αS = 0.
€
Fig(X):= x,y,z( ) ∈ R3
X ∧P x, y, z( ) = 0
$
%
&
'
(
)
.
We
compute
a
Gröbner
basis
of
those
equaQons
to
draw
the
figure.
αS ∈ R[x,y,z]
The
figure
Fig(X)
of
an
object
X
in
R3
X ∈ R[E] X ∈ R[W]or

18. 18
VisualizaQon
(1)

19. 19
VisualizaQon
(2)

20. Algebra
Symbolic
ComputaQons
in
CGA

21. Fig(X1)
=
Fig(X2)
?
21
X1 = w01 + w02 + w03 + w0∞ − w12 − 2 w13 − 6 w1∞ − w23 −5 w2∞
−4 w3∞ + w012 + w013 + w01∞ + w023 + w02∞ + w03∞ + 2 w123
+6 w12∞ + 5 w13∞ + 6 w23∞ + w0123 + w012∞ + w013∞ + w023∞
−5 w123∞ + w0123∞,
X2 = w0 + w1 + 2 w2 + 3 w3 + 7 w∞
The
appearance
of
X1
and
X2
are
different,
but
the
figure
of
X1
and
X2
are
same
and
it
is
a
point
{(1,2,3)}.
Our
implemented
funcQon
CGAEquaQonCheck
can
check
the
equaliQes
of
figures.
We
can
check
the
equality
of
figures
Fig(X1)
and
Fig(X2)
using
symbolic
computaQons.

22. FormalizaQon
of
Origami
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12 P13
O0=FirstO;
O1=Ori[O0, Ori2[P3, P1], {1}, π];
O2=Ori[O1, Ori2[P4, P3], {3}, π];
O3=Ori[O2, Ori2[P1, P4], {5}, −π];
O4=Ori[O3, Ori2[P5, P1], {4, 10}, π];
O5 = Ori[O4, Ori2[P6, P5], {13}, 0];
O6 = Ori[O5, Ori2[P7, P5], {29}, 0];
O7=Ori[O6 , Ori3[p2line[P8, P5],
p2line[P9, P8], 2], {27}, π];
O8=Ori[O7, Ori3[p2line[P8, P5],
p2line[P10, P8], 2], {59}, π];
O9=Ori[O8, Ori2[P11, P1], {15}, π];
O10=Ori[O9, Ori1[P7, P6], {31}, π];
O11=Ori[O10, Ori2[P7, P6], {11}, −π];

23. Origami
Graph
O = (Π, 〜, ≻)
Superposi5on
rela5on
≻
A
pair
of
face’s
ID
for
two
faces
superposed
directly.
≻
={
(f10,f14)
,(f14,f48)
,
(f16,f17)
,(f18,f26)
,(f22,f10)
,
(f23,f31)
,(f26,f48)
,(f30,f22)
,
(f31,f30)
,(f38,f18)
,(f39,f55)
,
(f48,f16)
,(f49,f51)
,(f50,f23)
,
(f50,f39)
,(f51,f50)
,(f54,f38)
,
(f55,f54)
}
f10
f14
f18
f26
f22
f30
f23
f31
f38
f54
f39
f55
f48
f50
f49
f51
f16
f17
Face
set
Π
Π
=
{
f10
,
f14
,
f18
,
f26
,
f22
,
f30
,
f23
,
f31
,
f38
,
f54
,
f39
,
f55
,
f48
,
f50
,
f49
,
f51
,
f16
,
f17
}
f49
={
p2
,
p13
,
p12
,
p4
}
f50
={
p6
,
p11
,
p7 }
f23
={
p5
,
p10
,
p1
}
f17
={
p15
,
p2
,
p7
}
etc.
The
vertex
coordinates
sets
of
each
polygon.
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12 P13
Adjacency
rela5on
〜
The
adjacent
face’s
ID
numbers
sets
to
every
side.
〜
=
{〜10
,
〜14
,
〜18
,
〜26
,
〜22
,
〜
30
,
〜23
,
〜31
,
〜38
,
〜54
,
〜39
,
〜55
,
〜48
,
〜50
,
〜49
,
〜51
,
〜16
,
〜17
}
〜10
={
f14
,
f22
,
f16
}
〜14
={
f48
,
f30
,
f10
}
〜22
={
f10
,
f13
,
Φ
}
〜16
={
f10
,
f17
,
f18
}
etc.
Where,
Φ
is
empty
set.
f10
f14
f18
f26
f22
f30
f23
f31
f38
f54
f39
f55
f48
f50
f49
f51
f16
f17
23
We follow the E-Origami-
System developed by Ida et al.
Origami
data
structure

24. Huzita-Hattori Axioms (1)
24

25. Huzita-Hattori Axioms (2)
25

26. A
sequence
of
origami
func5ons
for
making
Kabuto.
26
P1
P2 P3
P4
P1
P2 P3
P4
P1
P2 P3
P4
P5P6
P1
P2 P3
P4
P5P6
P7
P1
P5P6
P7
P8
P1
P5P6
P7
P8
P9
P1
P5P6
P7
P8
P9
P10
P1
P5P6
P7
P8
P9
P10
P1
P5P6
P7
P8
P9
P10
P1
P5P6
P7
P8
P9
P10
P11
P1
P5P6
P7
P8
P9
P10
P11
P1
P5P6
P7
P8
P9
P10
P11
A
design
sheet
of
origami
is
wriJen
by
successive
applicaQons
of
origami
funcQons.
O0=FirstO;
O1=Ori[O0,
Ori2[P3,
P1],
{1},
π];
O2=Ori[O1,
Ori2[P4,
P3],
{3},
π];
O3=Ori[O2,
Ori2[P1,
P4],
{5},
−π];
O4=Ori[O3,
Ori2[P5,
P1],
{4,
10},
π];
O5
=
Ori[O4,
Ori2[P6,
P5],
{13},
0];
O6
=
Ori[O5,
Ori2[P7,
P5],
{29},
0];
O7=Ori[O6
,
Ori3[p2line[P8,
P5],
p2line[P9,
P8],
2],
{27},
π];
O8=Ori[O7,
Ori3[p2line[P8,
P5],
p2line[P10,
P8],
2],
{59},
π];
O9=Ori[O8,
Ori2[P11,
P1],
{15},
π];
O10=Ori[O9,
Ori1[P7,
P6],
{31},
π];
O11=Ori[O10,
Ori2[P7,
P6],
{11},
−π];
Design
Sheet
Formula
Drawing

27. A
sequence
of
origami
func5ons
for
making
Kabuto.
27
P1
P2 P3
P4
P1
P2 P3
P4
P1
P2 P3
P4
P5P6
P1
P2 P3
P4
P5P6
P7
P1
P5P6
P7
P8
P1
P5P6
P7
P8
P9
P1
P5P6
P7
P8
P9
P10
P1
P5P6
P7
P8
P9
P10
P1
P5P6
P7
P8
P9
P10
P1
P5P6
P7
P8
P9
P10
P11
P1
P5P6
P7
P8
P9
P10
P11
P1
P5P6
P7
P8
P9
P10
P11
A
design
sheet
of
origami
is
wriJen
by
successive
applicaQons
of
origami
funcQons.
O0=FirstO;
O1=Ori[O0,
Ori2[P3,
P1],
{1},
π];
O2=Ori[O1,
Ori2[P4,
P3],
{3},
π];
O3=Ori[O2,
Ori2[P1,
P4],
{5},
−π];
O4=Ori[O3,
Ori2[P5,
P1],
{4,
10},
π];
O5
=
Ori[O4,
Ori2[P6,
P5],
{13},
0];
O6
=
Ori[O5,
Ori2[P7,
P5],
{29},
0];
O7=Ori[O6
,
Ori3[p2line[P8,
P5],
p2line[P9,
P8],
2],
{27},
π];
O8=Ori[O7,
Ori3[p2line[P8,
P5],
p2line[P10,
P8],
2],
{59},
π];
O9=Ori[O8,
Ori2[P11,
P1],
{15},
π];
O10=Ori[O9,
Ori1[P7,
P6],
{31},
π];
O11=Ori[O10,
Ori2[P7,
P6],
{11},
−π];
Design
Sheet
Formula
Drawing

28. VisualizaQon
(Origami)

29. Visualization functions
29

30. 3D
Folds

31. Three-dimensional origami folds
31

32. 3D folding using CGA
32

33. Simple proof using CGA
33
R1 =
1
2
+
w13
2
−
a
4 2
w3∞
€
R2 =
1
2
+
w13
2
−
a
2 2
w3∞
€
R3 =
1
2
+
w13
2
−
3a
4 2
w3∞
R = R3 ∗ R2 ∗ R1

34. A Mathematica module for Conformal Geometric Algebra and Origami Folding
34
Conclusion
• We
implemented
funcQons
that
realize
CGA
operaQons
using
symbolic
computaQons
in
MathemaQca.
hJps://github.com/KyushuUniversityMathemaQcs/MathemaQcaCGA
• Our
module
includes
funcQons
for
the
geometric
product,
the
inner
and
outer
product
of
a
CGA
which
s
an
extension
of
R3
with
e0
and
e∞.
• We
draw
the
figure
of
an
element
in
R3
by
solving
an
equaQon
system
using
Gröbner
basis.
• We
implemented
funcQons
of
2D
origami
folding
following
the
2D
origami
system
introduced
by
Ida
et
al.
• We
implemented
funcQons
which
compute
a
CGA
element
corresponding
to
a
3D
origami
folding.
• We
presented
a
simple
proof
of
the
3D
origami
property
using
a
symbolic
computaQon
of
CGA
equaQons.

35. Future
Work
• To
implement
3D
origami
funcQons
completely.
We
need
to
extend
the
data
structure
for
3D
origami.
• To
create
a
database
of
design
sheets
of
origami
folds
using
a
formula
in
CGA.
• To
find
an
aJracQve
folding
moQon
defined
by
a
formula
in
CGA.
• To
implement
a
funcQon
for
judging
the
collision
of
faces
in
3D
origami
folds.
• To
invesQgate
mathemaQcal
properQes
in
CGA,
especially
checking
equaliQes
of
CGA
objects.

36. A Mathematica module for Conformal Geometric Algebra and Origami Folding
36
Reference
1) T.
Ida,
Huzita's
basic
origami
fold
in
geometric
algebra,
16th
InternaQonal
Symposium
on
Symbolic
and
Numeric
Algorithms
for
ScienQfic
CompuQng
(SYNASC
2014),
11-­‐13.
IEEE
computer
society,
2014.
2) T.Ida,
H.Takahashi,
M.Marin,
A.Kasem
and
F.Ghourabi,
ComputaQonal
origami
system
EOS.
Proc.
4th
InternaQonal
Conference
of
Origami
Science,
MathemaQcs
and
EducaQon
(4OSME),
pages
285-­‐-­‐293,
2009.
3) M.Kondo,
T.Matsuo,
Y.Mizoguchi,
and
H.Ochiai.
A
mathemaQca
module
for
conformal
geometric
algebra
and
origami
folding.
7th
InternaQonal
Symposium
on
Symbolic
ComputaQon
in
Souware
Science
(SCSS2016),
volume
39
of
EPiC
Series
in
CompuQng,
68-­‐80.
EasyChair,
2016.
4) C.Perwass.
Geometric
Algebra
with
ApplicaQons
in
Engineering,
volume
4
of
Geometry
and
CompuQng.
Springer,
2009.