New Math this week:  the main results this week involved the so-called Magic Theorems, that establish important results about the orbifold notation we have been learning to use to name symmetry groups.  To understand the theorems we need to introduce the basic symbols for the notation.

• $\large *$ Represents a mirror line
• X Represents a glide reflection
• $n\in \mathbb{N}$ represents a natural number $n > 1$, a rotation center of order $n$.
• $O$ represents a pattern without any of the above features.

To each of these symbols there is a cost associated.  By considering all possible valid strings consisting of these symbols, one can then associate to each string a cost by summing over the costs associated to the constituent symbols.  One obtains some beautiful theorems.  But first we need to introduce some further concepts.

The Euler characteristic of a surface.   Suppose I have a set of polygons with the property that every edge is shared by exactly two polygons.  This set of polygons is called a 2-complex and determines a two-dimensional surface without boundary. Consider 2-complexes which yield the 2-sphere as surface.  Let V, E, and F denote the number of vertices, edges, and faces, resp. in the 2-complex. Then we showed by considering the platonic solids that $V-E+F=2$ for these surfaces, and gave reasons to believe that this relation is always true.  We define the associated function for a 2-complex $c$: the Euler characteristic $\lambda(c) := V-E+F$.  For a 2-sphere, $\lambda(c) = 2$.  Considering the 1-holed torus we showed that $\lambda(c)=0$ and, in general, for n-holed tori, $\lambda(c) = 2-2n$.

Euler characteristic of orbifolds. By considering the orbifolds we have been studying, we were able to extend the notion of Euler characteristic to these spaces, too.  The key idea is to introduce fractional points and edges at points and edges which are fixed by some group element.  One determines the fractional weights so that the resulting weights add up to 1 wherever the stabilizer of the group is non-trivial.  That is, make sure that the statement: “The copies of the fundamental domain cover the plane (or sphere) without overlap” remains true.

The cost function.  The cost function is defined as follows:

• $k(*) = k(X) = 1$,
• $k(n) = \dfrac{n-1}{n}$ if it comes before any $*$ in the string, otherwise $k(n) = \dfrac{n-1}{2n}$.
• $k(O) = 0.

Example: Let $s = 235$.  Then $k(x) = \dfrac{1}{2}+ \dfrac{2}{3} + \dfrac{4}{5} = \dfrac{118}{60}$.

Magical theorem for spherical groups:  Let $s$ be a valid string consisting of the above symbols.  Then the cost function $k(s) < 2$ exactly when $s$ describes a spherical orbifold $\mathcal{O}$.  In this case, the Euler characteristic of $\mathcal{O}$ is $\lambda({\mathcal{O}}) = 2-k(s)$, and the symmetry group of  $\mathcal{O}$ has order $\dfrac{1}{\lambda({\mathcal{O}})}$.

Example:  Applied to $235$, one obtains that this orbifold has Euler characteristic $\dfrac{1}{60}$ and the symmetry group has order 60.

There is a similar, simpler theorem for wallpaper groups.

Magical theorem for spherical groups:  Let $s$ be a valid string consisting of the above symbols.  Then the cost function $k(s) = 2 $ exactly when $s$ describes a spherical orbifold $\mathcal{O}$.

Further research: Note that the theorem makes no claims regarding the order of the group, since they are all infinite groups.One important invariant of a wallpaper group is the order $O_T$ of the translation group within the full group.  Can you find a way to derive this invariant from the Conway-Thurston orbifold notation for the group?

Further reading:  See the book Symmetry of Things by John Conway, et. al., Chs. 3, 4, and 6 for details and proofs of all the above themes and claims.  The book can be found  in the math library “Apparat”.

Important scheduling decision: Today in class we determined that the presentations for the summer semester would be held during the  first two weeks Tuesday, April 1 – Friday, April 11, although lectures begin only on April 14.  This choice was proposed by Mr. Gunn, since Easter school vacations run from April 14-April 28 and he has plans for family vacation during this time.  It’s also likely that getting the presentations over before the semester lectures start will be healthy for everyone involved.

Important calendar dates this year.  Mr. Gunn can’t resist pointing out the following dates from this semester are … well if not significant at least interesting:

• 7.11.13   The last time three consecutive primes will occur in a date until next century.
• 5.12.13   A Pythagorean triple date.  This won’t happen again until 17.8.15 — Mr. Gunn believes.  Can anyone prove or disprove this?
• 11.12.13  Three consecutive whole numbers.  Won’t happen again until next century.

List of Projects.  Here is a list of possible projects from Mr. Gunn’s fantasy land. Also, take a look at last year’s projects to see how such an idea can be implemented.  The projects entitled “Creation of Escher Tilings”, “A 3D Kaleidoscope” and “Dynamical Wallpaper Patterns” could be reworked again this year as there remains quite a lot to do. Of course you’re encouraged to come up with your own project ideas also.  I would like to meet with each team before Christmas to discuss the state of your project! One possibility would be to use the tutorials next week for this purpose.  If there are no objections I’ll prepare a sign-up sheet to pass around on Thursday’s lecture. (12.12.13).

Wallpaper group resource: The following chart  contains a list of all 17 wallpaper groups including the standard fundamental domains and the generators used in the discretegroup WallpaperGroup class.

The 17 wallpaper groups

George Hart videos.  I encourage you to take a look at the mathematical videos from George Hart, an American mathematician who does lots of interesting stuff with symmetry groups.  In particular, check out the videos involving group mathematical sculpture constructions (“Geometric barn-raising”). For example, this one.  I think we should consider doing something like this as a class after we come back from Christmas and hanging the result in the ground floor of the math building (outside of the lecture hall 005, etc).

10.12 Mathematics. Mr. Gunn introduced the remaining spherical groups, the ones that do not preserve 2 antipodal points.  They are the symmetry groups of the platonic solids. To be precise, the groups and the platonic solids are:

• *233    tetrahedron
• *234    cube and octahedron
• *235    pentagon dodecahedron

There are also the rotational subgroups of these groups: 233, 234, and 235.

Mr. Gunn explained how the fundamental region for these groups can be arrived at by splitting an equilateral spherical triangle into 6 smaller right triangles with angles $\dfrac{\pi}{2},\dfrac{\pi}{3},\dfrac{\pi}n$ for $n\in \{3,4,5\}$.  As a result these groups are known as triangle groups.  Check out the triangle group application here, or access it in eclipse via “Referenced Libraries->discretegroupCGG.jar->discreteGroup.demo.TriangleGroupDemo.class” then right-click and “Run as …->Java application”.

3*2. Finally there is the group 3*2 which has three mutually perpendicular mirrors which split the sphere into 8 right-angled triangles; in the center of the triangle is a rotation center of order 3.  Assignment5 doesn’t work quite right with this group, you are encouraged to fix it and send Mr. Gunn the result.

Euler characteristic.  In discussing the platonic solids Mr. Gunn pointed out that the number of vertices V, the number of edges E, and the number of faces F in all the platonic solids obey the law $V-E+F=2$.  He showed that standard operations to insert faces, edges, or vertices also leave the value of $V-E+F$ invariant.  This value is called the Euler characteristic of the surface E(S).  It is a topological invariant of the surface (in this case the sphere).  By considering a torus rolled up out of a single quadrilateral, one was able to calculate that for the torus $E(S) = V-E+F=0$.  For a two-holed torus one can show (by folding it up from 1 octagon) that $E(S) = V-E+F=-2$.

In general, E(S) = 2$(1-g)$ where $g$ is the genus of the surface (the number of holes it has). Mr Gunn claimed but postponed proving that the Thurston-Conway notation for 2-orbifolds and 2-manifolds has a close connection to the Euler characteristic.  He indicated that if one takes the name of an orbifold $\mathcal{O}$ as a sequence of symbols, one can attach a “cost” function K(s) to each symbol $s$ in such a way that the sum $\Sigma$ of the costs of all the  symbols satisfies $E(\mathcal{O}) = 2-\Sigma$.  Of course one also has to define what is meant by the Euler characteristic of $\mathcal{O}$, since up til now we only know what it means for a sphere or other compact orientable surface. So that will happen on Thursday 12.12

Christmas ornament application. To demonstrate these groups, Mr. Gunn hooked up the beamer and fired up template.Assignment5 from the git repository.  There’s a blog post on this application here and an image album here. You’re encouraged to play with this application to get a feel for these spherical groups.

Note: The tutorials next week will meet in the PORTAL (Second floor, same end of the building as BMS).  We will have a demo of the facilities there, including the 3D printing and scanning area.

Assignments. The continuation of Assignment 4 is described in the previous post.  It’s due this Friday Dec. 6.   Assignment 5 — creation of a Christmas calendar image — is also described there.  It’s due on Dec. 13.  December 9: See this update.

03.12.13:  Mr. Gunn continued the guided tour of the planar symmetry groups by considering symmetry groups of the plane with 2 independent translations: the wallpaper groups.  In contrast to the frieze groups, they can have rotational symmetries of order greater than 2.

Wallpaper group application. You can  run a jReality wallpaper group application from your eclipse project by navigating in the Package Explorer to “Referenced Libraries->discretegroupCGG.jar->discreteGroup.wallpaper.WallpaperPluggedIn.class” and then using right-mouse click to choose “Run as … Java application”.   (Here is a Java webstart of the same application. But be careful — if you run it under Java 7, which is sometimes difficult to avoid with webstarts, make sure the left hand panel is hidden or it runs v e r y slowly.) This application allows you to experiment with all 17 wallpaper groups — see the “Wallpaper” menu and the left inspection panel.  Here’s a picture generated the group 244 with the automatic painting option (choose “Wallpaper->run” option):

Wallpaper group resource: The following chart  contains a list of all 17 wallpaper groups including the standard fundamental domains and the generators used in the discretegroup WallpaperGroup class.

The 17 wallpaper groups

More group theory. To obtain a good answer for Assignment 4 it’s useful to understand a bit more about the internal structure of the frieze groups.  It turns out the the translation subgroup plans an important role.  To understand this, we need to introduce some notation.

Definition: A subgroup $H$ of a group $G$ is normal if $gHg^{-1} = H \forall g \in G$. We write $H \trianglelefteq G$.

Theorem: The translation subgroup $T$ of a euclidean group $G$ is a normal subgroup.

Proof:  A general  isometry $g$ can be written as $g(\mathbf{X}) = A.\mathbf{x} + T$ where $A \in O(n)$ is an orthogonal transformation and $T \in \mathbb{R}^n$ is the vector of the translation. Then $g^{-1} = A^{-1}.\mathbf{x} + A^{-1}.T$ and writing out the composition $gSg^{-1}$ where $S$ is the translation $\mathbf{x} \rightarrow \mathbf{x} + S$ yields the result $gSg^{-1}(\mathbf{x})= \mathbf{x} + A^{-1}.S$, which is clearly a translation. \qed

Cosets.  A  subgroup $H \trianglelefteq G$ determines an equivalence relation on the group $G$ as follows: Define $\mathbf{x} \sim \mathbf{y} \iff x^{-1}y \in H$.  We leave it to the reader to verify that this is an equivalence relation.  Hence, it partitions $G$ into equivalence classes of elements.  Each equivalence class is called a left coset of $G$ with respect to $H$.  A similar definition involving $xy^{-1}$ yields the right cosets. The coset containing $id$ is clearly $H$ itself. [Exercise: when $H$ is normal, the left cosets and the right cosets coincide.] The coset containing $\mathbf{x} \in G$ is written $[\mathbf{x}]$.  [Exercise: Every element of $[\mathbf{x}]$ can be uniquely written as $\mathbf{x}\mathbf{h}_x$ for some $h_x \in H$.] Let us choose a set of representative ${\mathbf{a}_i}$ for the cosets.  Then $\mathbf{x} \in [\mathbf{a}_i]$ can be written as $\mathbf{x} = \mathbf{a}_i \mathbf{h}_x$ for some $\mathbf{h}_x \in H$.

The quotient group. The cosets of a normal subgroup have an induced group structure. Indeed, let $[\mathbf{x}]$ and $[\mathbf{y}]$ be two cosets.  Then define a product on cosets by $[x][y] = [xy]$. Using our canonical representatives we can write  the RHS as $[(\mathbf{a}_i \mathbf{h}_x) (\mathbf{a}_j \mathbf{h}_y)]$.  Now since right and left cosets are the same, there exists $\mathbf{h}^{‘}_x \in H$ such that $\mathbf{h}_x \mathbf{a}_j =   \mathbf{a}_j \mathbf{h}^{‘}_x$. Then moving parentheses $[\mathbf{a}_i (\mathbf{h}_x \mathbf{a}_j) \mathbf{h}_y] = [\mathbf{a}_i \mathbf{a}_j \mathbf{h}^{‘}_x \mathbf{h}_y] = [ \mathbf{a}_i \mathbf{a}_j]$ so the result is clearly independent of the choice of $\mathbf{x}$ and $\mathbf{y}$.

We write $G/H$ and call it the quotient group of $G$ by $H$.

Connection to Assignment 4. At the conclusion of thursday’s lecture, Mr. Gunn showed how these ideas could be applied to Assignment 4 to generate the desired list of elements of the freize groups. Here’s what the resulting code for updateCopies() looks like:

Matrix[] cosets = null;
Matrix generatingTranslation = null;
int numCosets = 0;
switch(whichGroup)	{
case 0:
	numCosets = 1;
	cosets = new Matrix[1];
	cosets[0] = new Matrix();
	generatingTranslation = MatrixBuilder.euclidean().translate(1, 0, 0).getMatrix();
	break;
case 1:
	numCosets = 2;
	cosets = new Matrix[numCosets];
	cosets[0] = new Matrix(); // identity
	cosets[1] = MatrixBuilder.euclidean().reflect(new double[]{1,0,0,0}).getMatrix();
	generatingTranslation = MatrixBuilder.euclidean().translate(2, 0, 0).getMatrix();
	break;
	// implement this group (* infinity infinity ) here as above in case 0
	// there are two vertical mirrors a distance 1 apart.
case 3:
	generatingTranslation = MatrixBuilder.euclidean().reflect(new double[]{0,1,0,0}).translate(1,0,0).getMatrix();
	numCosets = 1;
	cosets = new Matrix[1];
	cosets[0] = new Matrix();
	break;
case 6:
	// implement this group (* 2 2 infinity) here as above in case 0
	// there are two vertical mirrors a distance 1 apart and a horizontal mirror  in the line y = 0.
	generatingTranslation = MatrixBuilder.euclidean().translate(2,0,0).getMatrix();
	numCosets = 4;
	cosets = new Matrix[4];
	cosets[0] = new Matrix();
	cosets[1] = MatrixBuilder.euclidean().reflect(new double[]{1,0,0,0}).getMatrix();
	cosets[2] = MatrixBuilder.euclidean().reflect(new double[]{0,1,0,0}).getMatrix();
	cosets[3] = MatrixBuilder.euclidean().rotateZ(Math.PI).getMatrix();
	break;
        .... // other cases not included here
}
	int outerLoopCount = count/numCosets, counter = 0;
	// run through the powers of the generating translation ...
	for (int j = 0; j <= outerLoopCount; ++j) {
		Matrix powerOfTranslation = Matrix.power(generatingTranslation, (j - outerLoopCount/2));
		// and multiply it on the right by each of the coset representatives
		for (int k = 0; k<numCosets; ++k)	{ 			// skip over non-existing scene graph components at end 			if (counter >= count) continue;
			SceneGraphComponent child = translationListSGC.getChildComponent(counter++);
			Matrix tmp = new Matrix(cosets[k].getArray().clone());
			tmp.multiplyOnLeft(powerOfTranslation);
			tmp.assignTo(child);
		}
	}

}