Escher's Tessellations: The Symmetry of Wallpaper Patterns...
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Escher’s Tessellations: The Symmetry of Wallpaper Patterns III 3 February 2012 Symmetry III 3 February 2012 1/16
Transcript of Escher's Tessellations: The Symmetry of Wallpaper Patterns...
Escher’s Tessellations:The Symmetry of Wallpaper Patterns III
3 February 2012
Symmetry III 3 February 2012 1/16
In the past two classes we saw many examples of Escher’s tessellations,and many different combinations of symmetry. We will talk briefly aboutthe broad mathematical structures used to classify wallpaper patterns, andwe’ll see pictorial examples of the different symmetry types which arepossible.
Symmetry III 3 February 2012 2/16
To each wallpaper pattern one can consider the collection ofisometries which, when applied to the picture, superimposes it exactlyonto itself.
To understand a situation, mathematicians often look for some sort ofstructure on collections of objects rather than working just with theindividual object.
For example, the collection of numbers has the operation of addition,which takes two numbers and adds them, producing a third number.
Symmetry III 3 February 2012 3/16
To each wallpaper pattern one can consider the collection ofisometries which, when applied to the picture, superimposes it exactlyonto itself.
To understand a situation, mathematicians often look for some sort ofstructure on collections of objects rather than working just with theindividual object.
For example, the collection of numbers has the operation of addition,which takes two numbers and adds them, producing a third number.
Symmetry III 3 February 2012 3/16
To each wallpaper pattern one can consider the collection ofisometries which, when applied to the picture, superimposes it exactlyonto itself.
To understand a situation, mathematicians often look for some sort ofstructure on collections of objects rather than working just with theindividual object.
For example, the collection of numbers has the operation of addition,which takes two numbers and adds them, producing a third number.
Symmetry III 3 February 2012 3/16
The collection of isometries has the property that two isometries canbe combined, or composed, by performing one, then the other,producing a third isometry.
A glide reflection is an example of two isometries being composed.We get it by performing a reflection followed by a translation.
As we saw on Wednesday, if we compose a horizontal and a verticalreflection, we get a 180 degree rotation.
Symmetry III 3 February 2012 4/16
The collection of isometries has the property that two isometries canbe combined, or composed, by performing one, then the other,producing a third isometry.
A glide reflection is an example of two isometries being composed.We get it by performing a reflection followed by a translation.
As we saw on Wednesday, if we compose a horizontal and a verticalreflection, we get a 180 degree rotation.
Symmetry III 3 February 2012 4/16
The collection of isometries has the property that two isometries canbe combined, or composed, by performing one, then the other,producing a third isometry.
A glide reflection is an example of two isometries being composed.We get it by performing a reflection followed by a translation.
As we saw on Wednesday, if we compose a horizontal and a verticalreflection, we get a 180 degree rotation.
Symmetry III 3 February 2012 4/16
In arithmetic, addition satisfies:
The associative property - e.g., 3 + (2 + 5) = (3 + 2) + 5.
An identity 0 - e.g., 3 + 0 = 3.
Additive inverses - e.g., 8 + (−8) = 0.
Symmetry III 3 February 2012 5/16
In arithmetic, addition satisfies:
The associative property - e.g., 3 + (2 + 5) = (3 + 2) + 5.
An identity 0 - e.g., 3 + 0 = 3.
Additive inverses - e.g., 8 + (−8) = 0.
Symmetry III 3 February 2012 5/16
In arithmetic, addition satisfies:
The associative property - e.g., 3 + (2 + 5) = (3 + 2) + 5.
An identity 0 - e.g., 3 + 0 = 3.
Additive inverses - e.g., 8 + (−8) = 0.
Symmetry III 3 February 2012 5/16
In arithmetic, addition satisfies:
The associative property - e.g., 3 + (2 + 5) = (3 + 2) + 5.
An identity 0 - e.g., 3 + 0 = 3.
Additive inverses - e.g., 8 + (−8) = 0.
Symmetry III 3 February 2012 5/16
Composing isometries also satisfy the same three properties.
The analogue of 0 is the “no motion”, or identity, isometry. Also,each isometry has an inverse which, when performed after theoriginal, results in no motion at all.
For example, the inverse of a rotation by 90 degrees counterclockwiseis a rotation by 90 degrees clockwise. A reflection is its own inverse.That is, performing a reflection twice accomplishes the same thing asno motion at all.
Symmetry III 3 February 2012 6/16
Composing isometries also satisfy the same three properties.
The analogue of 0 is the “no motion”, or identity, isometry. Also,each isometry has an inverse which, when performed after theoriginal, results in no motion at all.
For example, the inverse of a rotation by 90 degrees counterclockwiseis a rotation by 90 degrees clockwise. A reflection is its own inverse.That is, performing a reflection twice accomplishes the same thing asno motion at all.
Symmetry III 3 February 2012 6/16
Composing isometries also satisfy the same three properties.
The analogue of 0 is the “no motion”, or identity, isometry. Also,each isometry has an inverse which, when performed after theoriginal, results in no motion at all.
For example, the inverse of a rotation by 90 degrees counterclockwiseis a rotation by 90 degrees clockwise. A reflection is its own inverse.That is, performing a reflection twice accomplishes the same thing asno motion at all.
Symmetry III 3 February 2012 6/16
Group Theory
Group theory studies collections of objects together with an operationwhich satisfies the same three properties mentioned above whichaddition satisfies.
Group theory originated in the early 19th century through the work ofGalois, who introduced the concept in order to study solutions ofpolynomial equations.
Symmetry III 3 February 2012 7/16
Group Theory
Group theory studies collections of objects together with an operationwhich satisfies the same three properties mentioned above whichaddition satisfies.
Group theory originated in the early 19th century through the work ofGalois, who introduced the concept in order to study solutions ofpolynomial equations.
Symmetry III 3 February 2012 7/16
The collection of isometries associated to a wallpaper pattern is agroup.
There is one important difference between the group of isometriesand the group of numbers with addition. The latter satisfies thecommutative property (e.g., 2 + 3 = 3 + 2), while the former doesnot.
To illustrate this, we consider a vertical reflection and a quarter turn(counterclockwise) rotation.
Symmetry III 3 February 2012 8/16
The collection of isometries associated to a wallpaper pattern is agroup.
There is one important difference between the group of isometriesand the group of numbers with addition. The latter satisfies thecommutative property (e.g., 2 + 3 = 3 + 2), while the former doesnot.
To illustrate this, we consider a vertical reflection and a quarter turn(counterclockwise) rotation.
Symmetry III 3 February 2012 8/16
The collection of isometries associated to a wallpaper pattern is agroup.
There is one important difference between the group of isometriesand the group of numbers with addition. The latter satisfies thecommutative property (e.g., 2 + 3 = 3 + 2), while the former doesnot.
To illustrate this, we consider a vertical reflection and a quarter turn(counterclockwise) rotation.
Symmetry III 3 February 2012 8/16
The series in blue results from doing a vertical reflection followed by a90 degree rotation. The series in red results from performing the twoisometries in the opposite order. Since the results are different, theorder in which isometries are performed matters.
Symmetry III 3 February 2012 9/16
The series in blue results from doing a vertical reflection followed by a90 degree rotation. The series in red results from performing the twoisometries in the opposite order. Since the results are different, theorder in which isometries are performed matters.
Symmetry III 3 February 2012 9/16
The series in blue results from doing a vertical reflection followed by a90 degree rotation. The series in red results from performing the twoisometries in the opposite order. Since the results are different, theorder in which isometries are performed matters.
Symmetry III 3 February 2012 9/16
It is through the study of groups of isometries that the classification of allpossible types of symmetry of wallpaper patterns was made.
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Classification of Wallpaper Patterns
It was discovered that there are exactly 17 different types ofsymmetry in wallpaper patterns, by seeing that there are 17 differentgroups of isometries. This was completed by Fedorov, Schoenflies,and Barlow at the end of the 19th century.
Escher discovered the classification on his own. He drew pictures for16 of the 17 symmetry types.
Symmetry III 3 February 2012 11/16
Classification of Wallpaper Patterns
It was discovered that there are exactly 17 different types ofsymmetry in wallpaper patterns, by seeing that there are 17 differentgroups of isometries. This was completed by Fedorov, Schoenflies,and Barlow at the end of the 19th century.
Escher discovered the classification on his own. He drew pictures for16 of the 17 symmetry types.
Symmetry III 3 February 2012 11/16
You can find a PDF file showing pictures of all 17 symmetry types,including Escher drawings for 16 of them, at the course website by clickingon Handouts, and then clicking on 17 Wallpapers.pdf.
You can also find a webpage, also available from the Handouts link, titledWallpaper Patterns.
We will look at the 17 symmetry types now.
Symmetry III 3 February 2012 12/16
Crystallographers, interested in the chemical properties of crystals,studied their symmetry, and in the late 19th century classified thetypes of symmetry of crystals. They found that there are 230 differentsymmetry types. This is the 3-dimensional analogue of theclassification of wallpaper patterns.
Group theory has been used in encryption, coding theory, quantummechanics, and crystallography, among other areas.
Symmetry III 3 February 2012 13/16
Crystallographers, interested in the chemical properties of crystals,studied their symmetry, and in the late 19th century classified thetypes of symmetry of crystals. They found that there are 230 differentsymmetry types. This is the 3-dimensional analogue of theclassification of wallpaper patterns.
Group theory has been used in encryption, coding theory, quantummechanics, and crystallography, among other areas.
Symmetry III 3 February 2012 13/16
Quiz Question
Which of these pictures has rotational symmetry?
A The left picture only
B The right picture only
C Both
D NeitherSymmetry III 3 February 2012 14/16
Next Week
We will discuss encoding data and encryption of data next week.Encryption is what allows us to do internet shopping, while encoding datain clever is necessary to allow CDs and DVDs to play correctly even ifthere is some dirt on the disk. This will appear to be a completelyunrelated topic to what we discussed this week, but there are commonmathematical ideas in both subjects.
Symmetry III 3 February 2012 15/16
Assignment due next Friday
Do one of the following three activities:
Variant 1: Draw a tessellation starting with a square (or rectangle) byfollowing the instructions in the link Tessellations from Squares. Also,determine if the resulting tessellation has rotational, reflectional, and/orglide reflectional symmetry.
Variant 2: Draw a tessellation starting with a triangle by following theinstructions in the link Tessellations from Triangles Also, determine if theresulting tessellation has rotational, reflectional, and/or glide reflectionalsymmetry.
Variant 3: Look over the pictures of the 17 symmetry types in 17Wallpapers.pdf. Give a plausible reason why Escher did not draw a picturefor the symmetry type on the third page of that handout (which isnumbered page 59), when he drew (often many) pictures for the other 16symmetry types? Give some rationale for your opinion.
Symmetry III 3 February 2012 16/16
