Symmetry in Shapes Theory and Practice

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Tutorial

Symmetry in Shapes Theory and Practice

Niloy Mitra Maksim Ovsjanikov Mark Pauly Michael Wand Duygu Ceylan

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Geometry

geo = earth metria = measure

γεωµετρία

“The branch of mathematics

concerned with questions of shape, size, relative position of figures, and the properties of space.”

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10

⁰

Charles and Ray Eames

Powers of Ten, 1977 10

^-9

10

^-8

10

^-2

10

²

10

³

10

⁵

10

⁷

10

²¹

10

²⁵

10

^-4

10

^-5

10

^-16

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Symmetry

συµµετρία

1. “similarity, correspondence, or balance among systems or parts of a system

2. “an exact correspondence in position or form about a given point, line, or plane”

3. “beauty or harmony of form based on a proportionate arrangement of parts”

Collins English Dictionary

source: wikipedia

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Symmetry

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Symmetry

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Symmetry

Group Theory

• Mathematical language of symmetry

H. Weyl, Symmetry. Princeton University Press, 1952

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Transformations

Scale

Translation Rotation

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Symmetry Groups

Symmetry as invariance to transformations

Rotation by

³⁶⁰^◦

5 = 72^◦ 2 · 360^◦

5 = 144^◦ 3 · 360^◦

5 = 216^◦ 4 · 360^◦

5 = 288^◦ 5 · 360^◦

5 = 360^◦ = 0^◦

Cyclic Group C

5

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Symmetry Groups

Symmetry as invariance to transformations

2 · 360^◦

5 = 144^◦ 3 · 360^◦

5 = 216^◦ 4 · 360^◦

5 = 288^◦ 5 · 360^◦

5 = 360^◦ = 0^◦

Reflection

Dihedral Group D

5

Cyclic Group C

5

Rotation by

³⁶⁰^◦

5 = 72^◦

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Symmetry Groups

Group Generators

Dihedral Group D5

= 3

= 4

= 5

= 2

= =

=

2

=

3

=

4

generating transformations

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Symmetry Groups

Group Axioms

•

Closure a, b ∈ G → a · b ∈ G

a = Ref. A b = Ref. B

= =

?

a ⋅ b = Ref. A ⋅ Ref. B = Rot. 288°

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Symmetry Groups

Group Axioms

•

Closure

•

Associative

a, b ∈ G → a · b ∈ G

a, b, c ∈ G → (a · b) · c = a · (b · c)

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Symmetry Groups

Group Axioms

•

Closure

•

Associative

•

Identity

a, b ∈ G → a · b ∈ G

a, b, c ∈ G → (a · b) · c = a · (b · c)

∃ 1 ∈ G → ∀ a ∈ G : 1 · a = a · 1 = a

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Rot. 72°

Symmetry Groups

Group Axioms

•

Closure

•

Associative

•

Identity

•

Inverse

a, b ∈ G → a · b ∈ G

a, b, c ∈ G → (a · b) · c = a · (b · c)

∃ 1 ∈ G → ∀ a ∈ G : 1 · a = a · 1 = a

∀ a ∈ G ∃ b → a · b = b · a = 1

Rot. 288°

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Symmetry Groups

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Symmetry Groups

Group Generators

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Patterns

1D - Frieze Groups 2D - Wallpaper Groups

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Human Brain

Symmetry Groups?

Spiral Galaxy Antibody

Design by F. Gehry Roof Construction Metal Foam

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Classification

Global vs. Partial

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Classification

Global vs. Partial

Exact vs. Approximate

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Classification

Global vs. Partial

Exact vs. Approximate

Intrinsic vs. Extrinsic

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Understanding Geometry

Physical Object

Digital Measurement

Acquisition Reconstruction

Geometry Representation

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Understanding Geometry

Symmetry Analysis Reconstruction

Symmetry encodes Redundancy

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Symmetry & Information

“100 Random Points” “A 10x10 Regular Grid of Points”

High Information Content Low Information Content

Symmetry is Absence of information

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Symmetry & Information

→ structure discovery by minimizing representation cost

Symmetry is Absence of information

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Symmetry & Information

→ structure discovery by minimizing representation cost