Talk:Math and the Art of M. C. Escher

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I'm changing my tune and agreeing that this really is the place to hold this sort of book-wide discussion, so I'm organizing it a bit. Bryan 22:54, 7 September 2007 (CDT)

Projects on the main page

I created a bunch of new art projects this semester, and so far have put them on the main page, with the others. But it's really clutttering up the place. Not sure how to resolve the issue - possibly we should have a whole third section with projects and field trips, or else a secondary page with a prominent link on the main page. I would probably lean towards the latter, which will make the contents page cleaner. I could make some distinctive links that say 'Explorations', 'Exercises', 'Projects'. Thoughts? Bryan 11:27, 1 May 2012 (CDT)

Need For a Non-content Area

I've noticed stuff on the wiki for information that is only useful for specific courses, for example Homework 1 and Diary Week 1. Even the course syllabus pages are, IMO, misplaced on this wiki - a generic "example syllabus" would be more appropriate. These things are clearly not part of the "textbook". I do feel like this project is intended to result in a publication quality piece of work, and adding stuff that will need to be cleaned later is detrimental to that goal. Also, as this text goes public, we will (hopefully) add new instructors at other institutions, and I don't believe we want them posting this sort of info.

I'd suggest that Steve move his diary to his personal user page User:Steve, and that other course-specific information be placed on a separate website entirely on euler, like most courses taught in the department.

Clearly, though, you guys really like using the wiki for this - it's certainly very convenient. Looking into the future, the department web site may move to a university run system that will be as easy to use as this wiki. If it isn't as easy as this wiki, then I have every intention of setting up a wiki for the department web site and you can put this sort of thing there.

In the short term, I could set up a "namespace", like i've done for the instructor information. You could precede these pages with the namespace name, then a colon, then use whatever name you want after that. For example, Instructor:Spherical Geometry Exploration Solutions. I'd suggest "Course" as the name for the namespace, so for example you'd do [[Course:Homework 1]].

Putting these things in a separate namespace means they are not treated as "content". Also, it'll make it much easier to find them later when they need to be destroyed.

Let me hear an amen, and I'll go ahead with the namespace.

I think it is appropriate to have links on the Main Page that go to the class-specific pages for whatever classes are currently active, as that makes it very handy for the students; we could go with a link that says "For current classes click here", but I would prefer not to layer the material under another link, if possible. In any case, it's important that the class-specific pages be included in the same Wiki context as the rest of this material.
I want my Diary links to appear in the same page as my course outline, as they are intended for the students to use (as well as myself); the user-page is not appropriate for that use. Steve 11:52, 10 September 2007 (CDT)

I agree that the homework assignments needs to be put on a separate place.

I'm not sure right now about the syllabi. It is not part of the main text of the book, but it would belong in the instructor section of the textbook. I don't think it follows that all instructors will put their syllabus online. I think it is however important that several sample syllabi be made available.We teach somewhat different versions of the course and having a handful of these versions available will be a nice starting point for others.

For now I would like the syllabi to stay where they are. It should be easy enough to move and / or rename them after the end of the semester (or no?)

--Barta 14:02, 9 September 2007 (CDT)

I certainly agree we need sample syllabi (well, in my dreams they're more like "pathways" that are graphically incorporated into the text). The ones you guys have up right now should stay on the main page - I wasn't suggesting they disappear somehow. By a 'non-content' area, I don't mean these pages are less linkable or less visible, they're just tagged as 'not part of the core text'. Unless you've got links to the syllabus page from outside, the namespace change should cause no problems. Do you think your students go directly to the syllabus page, or do they go to the main page and then click the link? Bryan 02:08, 10 September 2007 (CDT)

I think that they usually go to the main page and then click on the syllabus. But that can easily be changed next time I teach the course. I will make sure that new text related to my class all have the course designation.

This actually relates to something I have encountered a couple of times now. I have used several of Bryan's explorations (they are great). But I often find that I would have either added or deleted some questions or made some optional. There's no way I would want to see umpteen different versions of one lab appear, but having a slightly modified version under my own class heading might just be the solution I was looking for. I will experiment with that at some point.

Looking to the future: How do we think this will work when many people have their own courses and maybe adapted materials? Where would this be stored? Could they open their own wikibook site (through wikimedia?) and create their own class site? Would they want to install their own version of Wikimedia? I can't see all of this coming from our server? (This is assuming others will join :) ) Not sure we have to answer this right now, but it may be something that will be an issue later. --Barta 09:09, 11 September 2007 (CDT)

Stability of exercise numbers

I was looking at Sample Homework Assignments, Homework 1 - Symmetry, Homework 1, and thinking about your course syllabus pages. It seems like it would be helpful for new instructors to have sample course syllabi, and we've got a lot in that direction. I don't see any problem with the reading material, explorations, and other info. However, the homework assignments are problematic. Anneke has felt the need to cut-paste problems from the "book" into special homework/sample homework pages. I find this abhorrent - information should appear once, because otherwise it gets out of sync. What good will pages like that do for new and future instructors if they don't correspond to problems actually listed on the Exercises pages? I understand the motivation, though (especially if you look at the associated Talk pages): the problem numbers might change.

I think we have a clear pact that new problems should be added to the end of exercise pages during the semester, so at least your in-class assignments are stable. But I was expecting problem numbers might change during break, so that related problems can be grouped rationally. This will potentially break "sample homework" assignments and also mean that when re-teaching the course the instructor will have to double check problem numbers, rather than re-using the same assignment. This problem happens with printed books but only once every five years.

I don't have any great ideas for how to deal with this issue. My gut feeling is that it's better to just assign problems by number. If you have new problems that aren't on the exercise pages already, then by all means add them at the end. After this year, the exercise numbers will settle down anyway, and any "sample homework" pages can be kept up to date with a fairly quick once-over every now and then. Bryan 22:38, 8 September 2007 (CDT)

The main reason I have for cutting and pasting the problems into the assignment is that I hand out the homework in paper form. I really do not like the idea of only referring to problem numbers in something I hand out. I feel that this assignment should be able to stand on its own so that the students can work on it even if they are away from their computer. For that reason I will always have the entire problem on the homework assignment.

When it comes to having sample problems on the wikipage: then I can see having only the problems listed. I get the impression that some of this goes back to the question of what should be on our class pages and what should be part of our wiki?

--Barta 08:47, 10 September 2007 (CDT)

Right.. I see this wiki as an attempt to write a textbook. In that sense, why not assign problems by number? You'd do it for a printed book. These students all have computer access all the time, and even so could easily print the problem page for later use if they need to. I never had complaints about that sort of thing last year - they seem perfectly comfortable with the wiki as a replacement for a "book"

But really, it does come down to the inclusion of content. Pages that don't belong in a textbook are pages that are going to need to be removed and/or revised later. I think they also send a signal to offsite visitors that this is not a general purpose text, but is specific to our university and our course. I ask myself, "does this belong in a textbook?" and pretty clearly the answer is no.

Anyway, read on for the new "Course" namespace. Bryan 12:52, 10 September 2007 (CDT)

I agree with the point that it comes down to content for the Wiki and that text specific to one's course should not be part of the main wiki. I think part of the problem is that I started using the wiki format to also do my class materials. That really should be separate from the main text, but I don't know where else to put it. I don't really want to change format right now. It is very nice to be able to use the wiki-machinery to create the related pages. Would it be hard to set up a parallel departmental wiki where we could move the course specific material to? I'm not saying we have to do it right now, but that seems to be the right way to go (and I remember Bryan mentioning this before).

For as far as the homework assignments go: sorry but I just hate the idea of just numbers. I have had problems in the past. It's just a style difference I guess, but I think I should be able to write my assignments the way I see fit.... --Barta 09:10, 11 September 2007 (CDT)

Course Namespace

You can now create pages that start with "Course:" to flag them as non-textbook content. For example, Diary Week 1 should move to Course: Steve's Diary Week 1 and Homework 1 should move to Course: Math 124 F07 Bart Homework 1. Other than the "Course:" designation, these pages are not any different from regular wiki pages.

You'll notice I'm also suggesting you get a lot more specific in your page names.

I'm not going to move any pages, since you're both in the thick of the semester. It shouldn't be much effort to do at this point, though, especially if you use "Course:" for new pages in the future. Bryan 12:52, 10 September 2007 (CDT)

Symmetry groups and tessellations

Is it true that the symmetry group of a tessellation, together with one particular shape, uniquely determines the tessellation up to isometry?

This is plainly true for rosettes and friezes, but in these cases the symmetry groups are easily understood: There's just one or two generators, and it's plain how to generate the rosette or the frieze just by using the generator(s) of the symmetry group on a single figure. But what about for wallpaper groups? I've no idea what the generators are for them, in most cases, and it's not at all evident that applying the generators (whatever they are) to a single figure will generate the tessellation.

This is, I think, the primary question about classification of tessellations: Does knowing the symmetry group tell you what essentially what the tessellation is? Given that two tessellations have the same symmetry group, does it follow that there's a mapping from a fundamental domain of one to a fundamental domain of the other that, extended by the symmetries, maps one tessellation isomorphically onto the other?

I strongly suspect the answer to this is "yes", but it's entirely clear to me how to prove it, given that these symmetry groups have significantly complicated relators. Even less clear is how to do the mapping between any two tesselations which have the same symmetry group, as it's not easy to find what a fundamental domain is.

Knowing the answer to this--and knowing how to find the fundamental domain--would make it easier to show how geometry really does inform our understanding of Escher works.

Steve 18:49, 28 August 2007 (CDT)

I think I now have a better grasp of what I'm asking and (maybe) how to prove it:

What I hope is true (and it seems that the Kali program makes use of it, so I guess it must be true!) is that having the symmetry group and a single tile determine a tessellation. Here's a way of nailing down the issues:

You really need to get Grunbaum and Shephard


  • a (planar) figure is a compact region of the plane which is equal to the closure of its interior and homeomorphic to the disk
  • a tessellation (of the plane) is a collection of figures such that
    • the union of the figures is the plane and
    • no two figures intersect save at their boundaries
these two are right on with G-S, except they'd call it a "tile" not a "figure". A "figure" to me connotes that it might have "markings" on it.
  • a regular tessellation is a tessellation such that all the figures are congruent to one another
No.. regular means it uses a single regular polygon tile.. and is tile-transitive.. and is edge-to-edge. "All tiles congruent" is shockingly weaker than you think. Better is "isohedral", which means every tile can be moved to every other via an isometry of the whole pattern.
  • a symmetry of a tessellation is an isometry of the plane mapping each figure of the tessellation to a figure
    • for a tessellation T, Sym(T) is the collection of all its symmetries, clearly a group
This may not be a strong enough condition, but it sounds ok.


Let T be a regular tessellation of the plane. Then Sym(T) is transitive on T, i.e., for any two figures F and F' in T, there is an element g of Sym(T) such that g(F) = F'.

That's way false, using your def. of regular. It's even false for isohedral tilings.

If this is true, then I can believe the following: Given two tessellations T and T' with Sym(T) = Sym(T'), let f be any homeomorphism from a figure F in T to a figure F' in T'. Then f generates a global isomorphism from T to T', i.e., a mapping f* from the plane to the plane which carries any figure in T to a figure in T', in a manner that is covariant with respect to the actions of the symmetry groups.

I'm sure that's false, too. Bryan 17:02, 30 August 2007 (CDT)

Steve 16:43, 30 August 2007 (CDT)

Say two patterns have the same symmetry group. Then there's a linear affine transformation that conjugates the group of one to the group of the other. Not sure exactly how this is proved, though. For most of the groups, this transformation must be an isometry. Not p1, though, for example.

So, you can take your pattern and map it to the other, and this produces a linear transformation that takes any fundamental domain of one to a fundamental domain for the other. But it's not going to identify just any pair of FD's, like the tile used in A with the tile used in B.

So, this says almost nothing about "tessellations". They're vastly more complicated. Escher's "classification" is hardly that, merely a list of a few things he used. According to Grunbaum and Shephard (you should get a copy of this bible), there are 81 types of isohedral tilings, and 91 types if you mark (ie. draw on) the tile. That's about the simplest case - one tile, tile-transitive. We're not even talking about colors yet - there's already 46 symmetry groups, allowing color permutations, for a 2-color pattern.

The first question, though: does knowing the group and a tile determine the tessellation? It doesn't, but in practice it might often be enough. However: Take some shape (like a pentomino or something) that you can build a rectangle out of so the rectangle has no symmetry. Build another rectangle from the same shape. Now both rectangles tile with a p1 symmetry, the tesselations both use the same tile, but they're certainly different. There's certainly more convincing examples, but that's what I've come up with right now.

PS - Steve, we had our first "Edit conflict", as we were both working on this page at the same time. Exciting. I hope I didn't screw it up. Bryan 16:52, 30 August 2007 (CDT)

  • "No.. regular means it uses a single regular polygon tile.. and is tile-transitive.. and is edge-to-edge."
    • Well, I just needed a label for the concept I had in mind. Choose another label. Point is, I want to include "Escherizations" of the tessellations we do with polygons, all of which are congruent and meet vertex-to-vertex. I omitted vertex-to-vertex in what I called "regular" tessellations above; we put that in by specifying a finite number vertices (arbitrarily, really) on the boundary of the figure and demanding that where two figures intersect, any vertex of one must also be a vertex of the other. (Hmm, that sounds really weak; is that going to accomplish anything?)
  • "It's even false for isohedral tilings."
    • Eh? Isohedral is exactly what I'm looking for! "Every tile can be moved to every other via an isometry of the whole pattern" says precisely that the symmetry goup is transitive on figures.
  • "So, you can take your pattern and map it to the other, and this produces a linear transformation that takes any fundamental domain of one to a fundamental domain for the other. But it's not going to identify just any pair of FD's, like the tile used in A with the tile used in B."
    • Huh? If one can find a transformation that takes any fundamental domain of one to a fundamental domain of the other, how is that not precisely identifying any pair of FDs?

More generally: Where do the 17 wallpaper groups come from?? What is that the classification for? And how does Kali work, if not precisely by what I was indicating above, i.e., specifying a single figure and a group, then generating something in the plane from that (though not necessarily a tessellation)?

(And what happened to Tess? Is that still available?)

Steve 15:12, 4 September 2007 (CDT)


I changed the fieldtrip entries. Bryan had mentioned that we do not really want a list of individual fieldtrips on the main page. I left the link to the main fieldtrip page called "Seeking Symmetry" and then added my fieldtrip assignment to that page. It may be nice to have a list of fieldtrips and the associated assignments on that page. Hopefully others will add versions in the future (even in other cities we hope). --Barta 11:55, 6 September 2007 (CDT)

Edge-to-edge tessellations

Much of the time we want to consider only tessellations by polygons in which vertices meet only vertices; but other times we want to consider tessellations by polygons without that restriction (such as the brick pattern). How do we distinguish the two notions?

I admit that in class I often finesse (ie ignore) this subtlety. The term to use is "edge to edge" for the first kind.Bryan 22:43, 7 September 2007 (CDT)
"Edge to edge"? That's a very bad way to say "vertext to vertex"; the standard brick pattern is clearly putting edges to edges. I think I'll just have to say "vertex to vertex" in class. Steve 11:41, 10 September 2007 (CDT)
I see your point, but I didn't make up the terminology.. it's standard. Looking at Grunbaum-Shepard, I'm guessing maybe it has to do with the distinction between vertex/edge of tiles and vertex/edge of tessellation. Vertex-to-vertex sounds just as good to me.. on the other hand, in the standard brick pattern, every vertex of a tile is also the vertex of a neighboring tile, but the same's not true vis a vis edges. In fact, maybe it's just for ease of definition. A tiling is edge-to-edge if every edge of every tile is also a complete edge of a neighboring tile. A tiling is vertex-to-vertex if....what? Every vertex of every tile is surrounded by...? Is not on an edge...? Well, we're splitting hairs.Bryan 13:12, 10 September 2007 (CDT)
My meaning of "vertex to vertex" is "a vertex on one tile intersects another tile only in a vertex". I see the intended meaning of "edge to edge" is "an entire edge on one tile intersects another tile only in an entire edge"; I guess I can see how that usage would come into being, but what it really should be called is "entire edge to entire edge" (else the brick pattern fits: edge goes to union of pieces of edge). Steve 12:33, 11 September 2007 (CDT)

Fundamental Domain

I note that what is called the Fundamental Domain for some of the Escher drawings is not at all what I would call such. For instance, consider the yellow and black winged lions (in , the Symmetries gallery, it's in the 4th column, 8th one down): It's based on a rectangluar pattern, but it's certainly not got the symmetries of the standard rectangular pattern (pmm); instead, it's pgg. In the exposition here, I've seen similar patterns described as having a Fundmental Domain of a rectangle (drawing a rectangle around one or two of the lions and saying that the pattern is derived from that). I would, instead, say the FD is a single lion, as the pattern is created by replicating that tile by the pgg group of isometries (generators: a translation and a glide reflection). Steve 17:25, 6 September 2007 (CDT)

The term gets used in two ways: As the fundamental domain for the symmetry group itself, and as the fundamental domain for the subgroup of translations. I'm pretty sure we avoid the issue by never mentioning fundamental domains in this text, but obviously that's something to be careful about.


We really need to incorporate an index into this text: A student wants to know how regular tesselation is defined; where is that done?

How difficult is this to manage? Steve 11:55, 10 September 2007 (CDT)

Not sure, but it's not going to be trivial. The plan was to put terms in the "define" template, in the hopes that that will help with creating an index someday. I'll look into the technical side of what needs to be done. There might be a utility for this, or maybe it'll involve a little programming. For now, keep on using "define". At least there's a "search" box, although I have to admit "regular tessellation" was not a very successful search when I tried it.Bryan 13:22, 10 September 2007 (CDT)
What would be Really Useful would be a utility that allowed you to highlight a phrase and then click "Insert this into index". This would insert the phrase, alphabetically, into the index list and create a link back to the highlighted instance. Steve 12:37, 11 September 2007 (CDT)

Content of discussion pages

I have put some comments about some of the explorations and exercises on the discussion pages. I have created a template {{Instructor Notes}} for this. This template puts a subsection with the heading "Instructor Notes" into the page. It allows us to quickly add some comments that may be useful for other instructors. I had in mind pretty specific comments like "I often assign problem 6 because I use the results later in my course" or "watch out for question 4, it's a bit of a trick question". I hope people will find this helpful in the future.

It's probably not appropriate to put complete answers up there because the page is accessible to students. If they read it it may actually help them, because it will point out that they need to be on their toes :) --Barta 15:40, 17 September 2007 (CDT)

I like it, and the template.Bryan 20:43, 18 September 2007 (CDT)
Great idea! I have several notes I want to put up about difficult problems, one problem being based on another, etc. Steve 04:09, 19 September 2007 (CDT)


I added a statement that even though we consider ourselves as the copyright owners of this site, we have no problem with others using these materials for educational purposes. I added in something that as long as it's for educational purposes, people do not need to ask for permission. Does that sound about right to you guys?

If this becoems a more collaborative effort, can we even claim copyright? I already added Steve's name as an important contributor. His comments and additions have already improved the site a lot I think. But I wouldn't want to have to create a list of "important contributors" at some point (where's the cut-off between important contributions and some minor contributions for instance?). We have the community portal which allows us to give some credit ... --Barta 11:27, 9 October 2007 (CDT)

I've moved this to it's own page, and there's now a prominent link at the top of the main page, also at the bottom of every page. For now, I think we're fine claiming copyright, since it's almost entirely our work. We can change if we get a few other editors helping out. Bryan 03:15, 20 January 2009 (CST)