the symmetry of wallpaper - university of minnesotadavidm/docs/symmetry-of-wallpaper.pdf · david...

The symmetry of wallpaper

David Morawski

University of Minnesota

Math Club, University of Minnesota25 March, 2011

“Symmetry is everywhere”

In art:

The Alhambra, Spain

David Morawski (UMN) The symmetry of wallpaper 25 March, 2011 2 / 35

In architecture:

Apartment building, Berkeley, CA

In nature:

A fine looking cactus.David Morawski (UMN) The symmetry of wallpaper 25 March, 2011 4 / 35

And so on. . .

Types of transformations

What symmetries do you see?

Types of transformations: reflection

Identical patterns

These patterns are equivalent:

Identical patterns

Notation: reflections (*)

This pattern is denoted by the signature *632.

* denotes a reflection.

Order of numbers does not matter: *632 = *326 = *263 = . . .

Another pattern

Notation: rotations

This pattern has signature 3*3.

Notation: rotations

Another example

This pattern has signature 333.

Another example

Another example: two reflections?

This pattern has signature **.

Similar, but different. . .

Notation: glide reflection (x)

This pattern has signature *x.

Another glide reflection example

This pattern has signature xx.

What type of transformations are seen in this pattern?

This pattern has signature o (the “empty transformation”).

What type of transformations are seen in this pattern?

This pattern has signature o (the “empty transformation”).

The four fundamental transformations (1/2)

Figure: Reflection (*ABC. . . D)

Figure: Rotation (ABC. . . D)

Figure: Reflection (*ABC. . . D)

Figure: Rotation (ABC. . . D)

Figure: Glide reflection (x)

Figure: Translation (o)

Figure: Glide reflection (x)

Figure: Translation (o)

These four transformations are called isometries.

An isometry is a transformation of the plane that preserves distance.

These four isometries are the only isometries of the Euclidean plane.

Thus, they are the only transformations of our wallpaper patterns we need toand can consider.

Assigning costs to symbols. . .

Symbol Cost ($)o 22 1

...N N−1

Symbol Cost ($)* or x 12 1

...N N−1

*632 costs $1 + 512 + 1

3 + 14 = $2.

3*3 costs $ 23 + 1 + 1

3 = $2.

** costs $1 + 1 = $2.

*x costs $1 + 1 = $2.

...N N−1

*632 costs $1 + 512 + 1

3 + 14 = $2.

3*3 costs $ 23 + 1 + 1

3 = $2.

** costs $1 + 1 = $2.

*x costs $1 + 1 = $2.

...N N−1

*632 costs $1 + 512 + 1

3 + 14 = $2.

3*3 costs $ 23 + 1 + 1

3 = $2.

** costs $1 + 1 = $2.

*x costs $1 + 1 = $2.

...N N−1

*632 costs $1 + 512 + 1

3 + 14 = $2.

3*3 costs $ 23 + 1 + 1

3 = $2.

** costs $1 + 1 = $2.

*x costs $1 + 1 = $2.

...N N−1

*632 costs $1 + 512 + 1

3 + 14 = $2.

3*3 costs $ 23 + 1 + 1

3 = $2.

** costs $1 + 1 = $2.

*x costs $1 + 1 = $2.

The $2 Theorem

Theorem

The signatures of plane repeating patterns are precisely those with total cost $2.

A consequence of this is that there are precisely 17 distinct wallpaperpatterns.

This theorem will help us determine the signature of a pattern:

If we have yet to reach $2, we need to find more transformations.

When our signatures total to $2, we have found the pattern’s signature!

The $2 Theorem

Theorem

The $2 Theorem

Theorem

The $2 Theorem

Theorem

The $2 Theorem

Theorem

The $2 Theorem

Theorem

Identifying signatures

1 First mark any reflections.

2 Then find rotations.

3 Are there any glide reflections? These should also be distinct fromreflections.

4 If you’ve found none of the above, there must only be a translation!

Remember. . .

1 Rotation points should not lie on reflection lines.

2 Glide reflections should not cross reflection lines.

Remember. . .

Identifying signatures: Example 1

This pattern has signature 22x.

This pattern has signature 22*.

Identifying signatures around town: Example 1

This pattern also has signature 2*22.

Example 1 vs. Example 2

These brick walls have the same symmetry.

Example 1 vs. Example 2

These brick walls have the same symmetry.

Why only 17 types?

Let’s see why the $2 Theorem might imply there are only 17 signatures of planerepeating patterns:

Case 1: the signature is all blue (ABC. . . D).

Case 2: the signature is all red and no crosses (*ABC. . . D).

Case 3: the signature contains both red and blue or it contains an x.

Why only 17 types?

Case 1: all blue

We know that the string ABC. . . D costs $2.

If there’s an o in the signature, then the signature is just o since this symbolcosts $2.

Otherwise, there must be more than two symbols in ABC. . . D, since theyeach cost less than $1.

If there are exactly three symbols, then the signature can only be 632, 442,or 333.

If there are more than three symbols, the signature must be 2222 since eachsymbol costs at least $ 1

Case 1: all blue

Case 2: all red and no x

Note that *ABC. . . D costs $2 if and only if ABC. . . D does:

$1 +A− 1

2A+ · · ·+ N − 1

2N= $2 ⇐⇒ $

A− 1

A+ · · ·+ N − 1

Thus, the possible signatures in this case are *632, *442, *333, *2222, or **.

Case 2: all red and no x

Note that *ABC. . . D costs $2 if and only if ABC. . . D does:

$1 +A− 1

2A+ · · ·+ N − 1

2N= $2 ⇐⇒ $

A− 1

A+ · · ·+ N − 1

Thus, the possible signatures in this case are *632, *442, *333, *2222, or **.

Case 3: hybrid types

Note that given a signature with either red and blue symbols or an x, we canmake the following cost-preserving substitutions:

replace *nn with n*replace a final * with x

And these can be reversed by

replacing n* with *nnreplacing a x with *

So, we can use the signatures *632, *442, *333, *2222, or ** to determine allpossible hybrid signatures of $2 value:

*632 *442 *333 *2222 **↓ ↓ ↓ ↓

4*2 3*3 2*22 *x↓ ↓

22* xx↓

replace *nn with n*

replace a final * with x

*632 *442 *333 *2222 **↓ ↓ ↓ ↓

4*2 3*3 2*22 *x↓ ↓

22* xx↓

*632 *442 *333 *2222 **↓ ↓ ↓ ↓

4*2 3*3 2*22 *x↓ ↓

22* xx↓

replacing n* with *nn

replacing a x with *

*632 *442 *333 *2222 **↓ ↓ ↓ ↓

4*2 3*3 2*22 *x↓ ↓

22* xx↓

*632 *442 *333 *2222 **↓ ↓ ↓ ↓

4*2 3*3 2*22 *x↓ ↓

22* xx↓

*632 *442 *333 *2222 **↓ ↓ ↓ ↓

4*2 3*3 2*22 *x↓ ↓

22* xx↓

Conclusion: only 17 signatures!

So, we have

5 blue signatures

5 red signatures

7 hybrid signatures: 4*2, 3*3, 2*2, *x, 22*, xx, and 22x.

Thank you!

For more information, check out The Symmetries of Things by John HortonConway – an excellent, accessible read for mathematicians andnon-mathematicians alike.

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