Imagifractalous! 4: Fractal Binary Trees II

Now that the paper Nick and I wrote on binary trees was accepted for Bridges 2017 (yay!), I’d like to say a little more about what we discovered.  I’ll presume you’ve already read the first Imagifractalous! post on binary trees (see Day077 for a refresher if you need it).

Recall that in that post, I discussed creating binary trees with branching ratios which were 1 or larger.  Below are three examples of binary trees, with branching ratios less that 1, equal to 1, and larger than 1, respectively.

2016-12-18threetrees.png

It was Nick’s insight to consider the following question:  how are trees with branching ratio r related to those with branching ratio 1/r?  He had done a lot of exploring with graphics in Python, and observed that there was definitely some relationship.

Let’s look at an example.  The red tree is a binary tree with branching ratio r less than one, and the gray tree has a branching ratio which is the reciprocal r.  Both are drawn to the same depth.

2016-12-03-doubletree1.png

Of course you notice what’s happening — the leaves of the trees are overlapping!  This was happening so frequently, it just couldn’t be coincidence.  Here is another example.

tree1

Notice how three copies of the trees with branching ratio less than one are covering some of the leaves of a tree with the reciprocal ratio.

Now if you’ve ever created your own binary trees, you’ll likely have noticed that I left out a particularly important piece of information:  the size of the trunks of the trees.  You can imagine that if the sizes of the trunks of the r trees and the 1/r trees were not precisely related, you wouldn’t have the nice overlap.

Here is a figure taken from our paper which explains just how to find the correct relationship between the trunk sizes.  It illustrates the main idea which we used to rigorously prove just about everything we observed about these reciprocal trees.

tree2

Let’s take a look at what’s happening.  The thick, black tree has a branching ratio of 5/8, and a branching angle of 25°.  The thick, black path going from O to P is created by following the sequence of instructions RRRLL (and so the tree is rendered to a depth of 5).

Now make a symmetric path (thick, gray, dashed) starting at P and going to O.  If we start at P with the same trunk length we started with at O, and follow the exact same instructions, we have to end up back at O.

The trick is to now look at this gray path backwards, starting from O.  The branches now get larger each time, by a factor of 8/5 (since they were getting smaller by a factor of 5/8 when going in the opposite direction).  The size of the trunk, you can readily see, is the length of the last branch drawn in following the black path from O to P.  This must be (5/8)5 times the length of the trunk, since the tree is of depth 5.

The sequence of instructions needed to follow this gray path is RRLLL.  It turns out this is easy to predict from the geometry.  Recall that beginning at P, we followed the instructions RRRLL along the gray path to get to O.  When we reverse this path and go from O to P, we follow the instructions in reverse — except that in going in the reverse direction, what was previously a left turn becomes a right turn, and vice versa.

So all we need to do to get the reverse instructions is to reverse the string RRRLL to get LLRRR, and then change the L‘s to R‘s and the R‘s to L‘s, yielding RRLLL.

There’s one important detail to address:  the fact that the black tree with branching ratio 5/8 is rotated by 25° to make everything work out.  Again, this is easy to see from the geometry of the figure.  Look at the thick gray path for a moment.  Since following the instructions RRLLL means that in total, you make one more left turn than you do right turns, the last branch of the path must be oriented 25° to the left of your starting orientation (which was vertical).  This tells you precisely how much you need to rotate the black tree to make the two paths have the same starting and ending points.

Of course one example does not make a proof — but in fact all the important ideas are contained in this one illustration.  It is not difficult to make the argument more general, and we have successfully accomplished that (though this blog is not the place for it!).

If you look carefully at the diagram, you’ll count that there are exactly 10 leaves in common with these two trees with reciprocal branching ratios.  There is some nice combinatorics going on here, which is again easy to explain from the geometry.

You can see that these common leaves (illustrated with small, black dots) are at the ends of gray branches which are oriented 25° from the vertical.  Recall that this specific angle came from the fact that there was one more L than there were R‘s in the string RRLLL.

Now if you have a sequence of 5 instructions, the only way to have exactly one more L than R‘s is to have precisely three L‘s (and hence two R‘s).  And the number of ways to have three L‘s in a string of length 5 is just

\displaystyle{5\choose3}=10.

Again, these observations are easy to generalize and prove rigorously.

And where does this take us?

canopies.png

On the right are 12 copies of a tree with a braching ratio of r less than one and a branching angle of 30°, and on the left are 12 copies of a tree with a reciprocal branching ratio of 1/r, also with a branching angle of 30°.  All are drawn to depth 4, and the trunks are appropriately scaled as previously discussed.

These sets of trees produce exactly the same leaves!  We called this the Dual Tree Theorem, which was the culmination of all these observations.  Here is an illustration with both sets of trees on top of each other.

2016-12-14gtree.png

As intriguing as this discovery was, it was only the beginning of a much broader and deeper exploration into the fractal world of binary trees.  I’ll continue a discussion of our adventures in the next installment of Imagifractalous!

Mathematics and Digital Art: Update 2

It’s been about a month since my first update, so it’s time for another status report on my second semester teaching Mathematics and Digital Art.  It really has been a wonderful semester so far!

Later we’ll look at some student work (like Collette’s iterated function system),

IFSCollette

but first, I’d like to talk about course content.

The main difference from last semester in terms of topics covered was including a unit on L-systems instead of polyhedra.  You might recall the reasons for this:  first, students didn’t really see a connection between the polyhedra unit and the rest of the course, and second, the little bit of exposure to L-systems (by way of project work) was well-received.

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I’ve talked a lot about L-systems on my blog, but as a brief refresher, here is the prototypical L-system, the Koch curve.  The scheme is to recursively follow the sequence of turtle graphics instructions

F  +60  F  +240  F  +60  F.

There is also an excellent pdf available online, The Algorithmic Beauty of Plants.  This is where I first learned about L-systems.  It is a beautifully illustrated book, and I am fortunate enough to own a physical copy which I bought several years ago.

Talking about L-systems is also a great way to introduce Processing, since I have routines for creating L-systems written in Python.  Up to this point, we’ve just explored changing parameters in the usual algorithm, but there will a deeper investigation later.

One main focus, however, was just seeing the fractal produced by the algorithm.  When working in the Sage environment, the system automatically produced a graphic with axes labeled, enabling you to see what fractal image you created.

In Processing, though, you need to specify your screen space ahead of time.  So if your image is drawn off-screen, well, you just won’t see it.  You have to do your own scaling and translating, which is sometimes not a trivial undertaking.

I also decided to introduce both finite and infinite geometric series in conjunction with L-systems.  This had two main applications.

First, we looked at the Sierpinski triangle.  Begin with any triangle, and take out the triangle formed by joining the midpoints of the sides.  Then repeat recursively, creating the Sierpinski triangle.

Sierp3

Now assume your original triangle had an area of 1, and calculate the area of all the triangles you removed.  Since the process is repeated infinitely, this sum is just an infinite geometric series.  Interestingly, the sum of this series is 1, meaning, in some sense, you’ve taken away all the area — but the Sierpinski triangle is still left over!  This illustrates an idea not usually encountered by students before:  infinite sets of points with no area.  Makes for a nice discussion.

Second, we looked at the Koch curve (and similarly defined curves).  Using a geometric sequence, you can look at the length of any iteration of the polygonal path drawn by the recursive algorithm.  And, as expected, these paths get longer each time, and their lengths tend to infinity as the number of iterations increases.  This is another nice way to involve geometric sequences and series.

We’ll be doing more with L-systems in the next few weeks, so I’ll finish this discussion on my next update.

A highlight of the past month was a visit by artist Stacy Speyer.

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I’ll Never Know, by Stacy Speyer.

Having worked with weaving and textiles for some time, Stacy has moved on to an investigation of polyhedral forms.

075-WovenGeoInstallVert

Stacy’s talk provided a wonderful insight into integrating mathematics and art in ways we did not study in class.  One of the goals of the Bridges papers presentations and the guest speakers is to do precisely this

She writes:

I’m now on a mission to share the fun of making geometric forms with others; I designed Cubes and Things, a 3D coloring book.  These easy-to-make paper constructions have patterns that can be colored which emphasize different kinds of symmetric properties of the polyhedra.  I bring this fun activity to schools and other groups in the form of Polyhedra Parties.  And whenever possible, I still work on making more geometric art and learning more about math.

Visit Stacy’s website to take a look at her book, and view many more examples of her stunning work!

Now we’ll take a look at a few more examples of student artwork.  These pieces were submitted for the assignment on iterated function systems.  Karla created a piece which reminded her of icicles or twinkling lights.

IFSKarla

Lainey thought her piece looked like a bolt lightning coming out of a wizard’s staff.

IFSLainey

And Peyton’s piece reminder her of flowers.

IFSPeyton

Finally, as I did last semester, I asked students for some mid-semester comments on how the course was going.  You can see the complete prompt on Day 19 of the course website.  Here are a few of the comments:

I like how it takes a subject that we are all required to take and creates a real, palpable output. Rather than some types of math, where everything is theoretical, it creates a clear chain of events with an even clearer consequence.

[A]fter seeing the kinds of art works there are that involve the kind of math and programming we use, it opened up a new world of artistic possibilities.

What I enjoy most about this course aside from it being small and very interactive in terms of doing labs and having all of our questions answered, is the fact that I would never thought I would be able to create images using programming or math let alone enjoying the satisfaction of the final product.

I was pleased to read these responses, as they suggest the course is fulfilling its intended purpose.  But there were also suggestions for improvement — there was a consensus that the math moved a bit too quickly.  When we start the discussion on number theory for analyzing the Koch curve next week, I’ll make sure to keep an eye on the pace.  I’ll let you know how it goes in my next update in April!

On Coding VIII: LaTeX II

Today, I’ll conclude my remarks about my passion for using LaTeX.  As I was writing the last installment of On Coding, I realized that I had more to say than would fit in just one post.

Yet another wonderful thing about LaTeX is how many mathematicians and scientists use it — and therefore write packages for it.  You can go to the Comprehensive TeX Archive Network and download packages which make Feynman diagrams for physics, molecular structures for chemistry, musical scores, and even crossword puzzles or chessboards!  There are literally thousands of packages available.  And like LaTeX, it’s all open source.  That is a feature which cannot be overstated.  Arguably the world’s best and most comprehensive computer typesetting platform is absolutely free.

The package I use most often is TikZ — it’s a really amazing graphics package written by Till Tantau.  You can do absolutely anything in TikZ, really.  One extremely important feature is that you can easily put mathematical symbols in any graphic.

tikz1

This is nice because any labels in your diagram will be in the same font as your text.  I always find it jarring when I’m reading a mathematics paper or book, and the diagrams are labelled in some other font.

There is so much more to say about TikZ.  I plan to talk about it in more detail in a future installment about computer graphics, so I’ll stop here and leave you with one more graphic made with TikZ.

latex2

Another package I use fairly often is the hyperref package.  This is especially useful when you’re creating some type of report which relies on information found on the web.  For example, when I request funding for a conference, I need to include a copy of the conference announcement.  So I create a hyperlink (in blue, though you can customize this) in the document which takes you to the announcement online when you click on it.

These hyperlinks can also be linked to other documents in the cloud, so you can have a “master” document which links to all the documents you need.  Now that I’m approaching 100 blog entries, I plan on making an index this way.  I’ll create a pdf (using LaTeX, of course) which lists posts by topic with brief descriptions as well as hyperlinks to the relevant blog posts.

On to the next LaTeX feature!  I learned about this one from a colleague (thanks, Noah!) when I was writing some notes on Taylor series for calculus.  I used it as a text when I taught calculus; the notes are about 100 pages long.

I wanted to share these notes with others, and the style of the notes was such that the exercises weren’t at the end of the sections, but interwoven with the text.  Students are supposed to do the exercises as they encounter them.

But for other calculus teachers, it was helpful to include solutions to the exercises.  The problem in creating a solutions manual was that if I ever edited the notes, I’d have to also edit the solutions manual in parallel.  I knew this was going to happen, since when I gave exams on this material, I added those problems as supplementary exercises to the text.

Enter the ifthen package in LaTeX.  I created an exercise environment, so that every time I included an exercise, I had a block which looked like this:

\begin{exercise}

{….the exercise….}

{….the solution….}

\end{exercise}

Think of this as an exercise function with two arguments:  the text of the exercise, and the text of the solution.

Then I created a boolean variable called teacheredition.  If this variable was true, the exercise function printed the solutions with each exercise.  If false, the solutions were omitted.  This control structure was made easy by some functions in the ifthen package.

And that’s all there was to it!  So every time I created an exercise, I added the solution right after it.  Of course the exercises were automatically numbered as well.  No separate solutions manual.  Everything was all in one place.  If you have ever had to deal with this type of issue before, you’ll immediately recognize how unbelievably useful the ability to do this is!

While not really features of LaTeX itself, there are now places in the cloud where you can work on LaTeX documents with others.  I’d like to talk about the one Nick and I are currently using, called ShareLaTeX.   This is an environment where you can create a project, and then share it with others so they can work on it, too.

So when Nick and I work on a paper together, we do it in ShareLaTeX.  It’s extremely convenient.  We can edit the paper on our own, but most often, we use ShareLaTeX when we’re working together.  Usually, we’re working on different parts of the paper — but when one of us has something we want the other to see, it’s easy to just scroll down (or up) in the document and look at what’s been done.

Also nice is that it’s easy to copy projects — so as we’re about to make a big change (like use different notation, or alter a fundamental definition), our protocol is to make a copy of the current project to work on, and then download the older version of the project (just in case the internet dies).

It’s wonderful to use.  And it actually really came in handy when Nick was working on his Bridges paper for last year.  His computer hard drive seriously crashed.  But since we were working on ShareLaTeX, the draft of his paper was unharmed.

I hope this is enough to convince you that it might be worthwhile to learn a little LaTeX!  I seriously don’t know what I’d do without it.  And — as it bears repeating — it’s all open source, available to anyone.  So, really, why isn’t the whole world using LaTeX?  That’s a mystery for another day….

What is…Spherical Geometry?

This week, we’ll look at another type of geometry, namely spherical geometry.  Quite simply, this is the geometry of a sphere.  Here, a sphere is a set of points equidistant from a given center.  In other words, throughout this post, you should imagine only a surface, with nothing inside it — think of the rind of an orange, without any of the slices.  I began to describe spherical geometry in my original post, What is a Geometry?, so I’ll briefly summarize the ideas in that post first.  It wouldn’t hurt to review it…

From a Euclidean standpoint, there are points on the sphere, but obviously no straight lines.  In spherical geometry, we define a Point to be a pair of antipodal points on the sphere, and a Line to be a great circle on the sphere.  This results in two nice theorems of spherical geometry:  any two distinct Lines determine (intersect in) a single Point, and any two distinct points determine a single Line.

This is a departure from Euclidean geometry, for this means that there are no parallel Lines in spherical geometry, since distinct Lines always intersect.  But there is something more going on here.

Consider the statement “Any two distinct Lines determine a single Point.”  Now perform the following simple replacement:  change the occurrence of “Line” to “Point,” and vice versa.  This gives the statement “Any two distinct Points determine a single Line,”  and is called the dual of the original statement.

Thus, we have the situation that some statement and its dual are both true.  Now if this is true of some set of statements in spherical geometry — that the dual of each statement is true — and we derive a new result from this set of statements, then the following remarkable thing happens.  Since the dual of any statement we used is true, then the dual of the new result must also be true!  Just replace each statement used to derive the new result with its dual, and you get the dual of the new result as a true statement as well.

This is the principle of duality in mathematics, and is a very important concept.  We will encounter it again when we investigate projective geometry.

Triangles and trigonometry are different on the sphere, as well.  A spherical triangle is composed of three arcs of great circles, as in the image below.

Fig31.png

If this were Earth, you could imagine starting at the North Pole, following 0 degrees of longitude to the Equator (shown in yellow), follow the Equator to 30 degrees east longitude, then follow this line of longitude back to the North Pole.

On the sphere, though, sides of a triangle are actually angles.  Sure, you could measure the length of the arcs given the radius of the sphere, but that’s not as useful.  Consider the choice of units on the Earth — kilometers or miles?  We don’t want important  geometrical results to depend on the choice of units.  So a side is specified by the angle subtended by the arc at the center of the sphere.  This makes sense since the sides are arcs of great circles, and the center of any great circle is the center of the sphere.

The angles between the sides are angles, too!  A great circle is just the intersection of a plane passing through the origin and the sphere.  So the angle between any two sides is defined to be the angle between the planes which contain them.

Really nothing from Euclidean trigonometry is valid on the sphere.  For example, if you look at the triangle above, you can see that the angles between the sides are 30, 90, and 90 degrees.  These angles add to 210!  In fact, the three angles of a triangle always add up to more than 180 degrees on a sphere.  (You may notice that the sides of this triangle are also 30, 90, and 90 degrees, but this is just a coincidence.  The sides and angles are usually different.)

If A, B, and C are the angles of a spherical triangle, it turns out that the area of the triangle is proportional to ABC – 180.  This means that the smaller a triangle is, the closer the angle sum is to 180 degrees.

sphtriangle

There is no Pythagorean Theorem on the sphere, either.  In addition, if a, b, and c are the sides opposite angles A, B, and C, respectively, then we have formulas like

\cos c=\cos a\cos b+\sin a\sin b\cos C

and

\cos C=-\cos A\cos B+\sin A\sin B\cos c.

One interesting consequence of the second of these formulas is this.  If you know the angles of a triangle, you can determine the sides.  You can’t do this in Euclidean trigonometry, since the triangles may be similar, but of different sizes.  In other words, there are no similar triangles on the sphere.  Spherical triangles are just congruent, or not. You can’t have two different triangles with the same angles.

For now, we’ve considered our sphere as being embedded in a Euclidean space.  The definition of this surface is easy:  just choose a point as the center of your sphere, and then find all points which are a given, fixed distance — the radius of the sphere — from that point.  Sounds easy enough.

But can you imagine a sphere without thinking of the three-dimensional space around it?  Or put another way, imagine you were a tiny ant, on a sphere of radius 1,000,000 km.  That’s over 150 times the radius of the Earth!  How would you know you were actually on the surface of a sphere?  If you were that small and the sphere were that large, it would seem awfully flat to you….

So how could you determine the sphere was curved?  This is a question for differential geometry, which among other things, is about the geometry of a surface without any reference to a space it’s embedded in.  This is called the surface’s intrinsic geometry.

As an example of looking at the intrinsic geometry of the sphere, consider Lines.  Now you can’t say they’re great circles any more, since this relies on thinking of a sphere as being embedded in three-dimensional space; in other words, its extrinsic geometry.  You need the concept of a geodesic — in other words, the idea of a “shortest path.”

So if you’re the ant crawling between two points on a sphere, and you wanted to take the shortest path, you would have to follow a great circle arc.  So it is possible to define Lines only using properties of the surface itself.  But the mathematics to do this is really extremely challenging.

Lots of new ideas here — but we’ve just scratched the surface of a study of spherical geometry.  You can see how very different spherical geometry is from both Euclidean and taxicab geometry.  Hopefully you’re well on your way to wrapping your head around our original question, What is a Geometry?….

Art Exhibition: Golden Section 2017

Yesterday, artists from the Golden Section of the Mathematical Association of America contributed to yet another art exhibition!  Each Spring, members of the MAA from Northern California, Nevada, and Hawaii attend a regional conference — this year, at Santa Clara University.

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Woven, by Nick Mendler.

Last year the event was held at the University of California, Davis, and Shirley Yap from California State University, East Bay organized a highly successful exhibit — what we believe to be the first art exhibition ever to be a part of a sectional MAA meeting.  I asked Shirley to say a few words about what motivated her to take on this task.

I exhibited an art piece at the Joint Mathematics Meetings in 2016. It was an interactive piece and I wanted to see how people would experiment with it.  So I just hung around the exhibit for a while and not only saw how people played with my piece, but how they observed other pieces. The kind of delight that came from people’s faces convinced me that the art was really drawing them to math in a way that was different from how I had seen before. Perhaps because one is expected to sit in front of art for a long time to contemplate it, people felt relaxed enough to enjoy it.  Whatever it was I saw, I knew that I wanted to share the experience with others outside of the JMM.

When we put a call for artists out on our Golden Section website, we didn’t get any responses. So I went through years of JMM art exhibit catalogs and looked up each artist to see if they lived in our section.  Then I just started emailing them individually to ask if they were interested in showing their work at a local exhibition.

This year, I offered to help Shirley with organizing the exhibition.  Given what was involved in the second year, I have a new appreciation for Shirley’s dedication to spreading the word about mathematical art.  Such events do not organize themselves — and we are all grateful Shirley took on this huge task to start a new Golden Section tradition.

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Red Mandala, by Frank Farris.

We didn’t have as many artists participate this year — but that’s part of the ebb and flow of yearly events like these.  But the quality has high, as was the enthusiasm of the artists.  Two of the artists this year were undergraduates — Nick Mendler from the Univiersity of San Francisco, and Juli Odomo from Santa Clara University.  I think of them as future organizers of sectional MAA art exhibits….

In the morning, we had the usual opening remarks and a series of excellent speakers.  The art exhibit took place in parallel with the Student Poster Sessions, which took place after lunch from 1:00-2:30.  This was followed by another series of talks.  You can see the full program here.

phil
Islamic 8-fold Fractal Flower (Median), by Phil Webster.

I asked the artists to say a few words about their experience about creating or exhibiting mathematical art.  Here a few remarks.

Frank Farris (see artwork above):

I love the idea that we’re entering a golden age of mathematical art. New tools become available all the time and a growing community is finding new creative ways to use them. Can’t wait to see what the next years will bring.

I believe the sentiment in Gwen’s quote resonates very strongly with many mathematical artists.

Gwen Fisher:

The thing that keeps bringing me back to bead weaving is mathematics. Of course, I love colors of glass beads and the way they sparkle, but mostly, I keep returning to my seed beads because I keep finding new ways to use and represent mathematical structures with them.

gwenfisher
Pixel Painting Number VI “Sunnyvale Boogie Woogie” by Gwen Fisher.

Nick Mendler (see artwork above):

Since my first sectional meeting last Spring, I’ve continued research into the questions that generated my first mathematical artwork over a year ago.
Recognizing that my projects and thoughts are the most rewarding when realized through an aesthetic process has been not only productive, but has been a fascinating source of guidance to new questions. That focusing on more elegant images brings about more elegant mathematics has been only too clear from the sessions I’ve attended so far; I’m looking forward to seeing and learning from more art pieces!

Interested in organizing an art exhibit in your section?  Since I helped Shirley with the organizational details this year, I can say a bit about what’s involved in putting together an exhibition.

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Intense discussion about mathematical art at the exhibition.

The first step is, clearly, finding artists who want to show their work.  It would be easier to get a student worker to do the search Shirley undertook — but don’t forget about the exhibitions at the Bridges conferences!  Here is a link to both JMM and Bridges galleries.  You can also contact the SIGMAA-ARTS and request that an email blast be sent to members.

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Symmetric Koch Curve I, by Vince Matsko.

As far as the submission process goes, that’s pretty standard.  While it is always nice to accept every submission, sometimes it just isn’t possible.  The works should have some real mathematical content, and be of good quality.

Since not all artists necessarily have business cards (especially student artists), I had the idea of making nametags for those who wanted one.  You can download this nametag template in LaTeX if you would like, then edit and print onto cardstock.  (Note:  WordPress would not let me upload a .tex document, so I saved it as an Open Office document.)

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A Fine Mesh We’re In, © dan bach 2016.

It is a good idea to have an assistant or a student helper in the exhibition venue during the conference.  Not all artists attended the meeting, and so brought in their work at various times during the day.

Shirley had the wonderful idea of arranging a dinner for contributing artists after the conference.  Last year we went to an excellent Thai restaurant, and this year, Frank Farris generously offered to host a pot luck dinner (he provided the lasagna) at his house.  These have been very wonderful events, and give artists the opportunity to get to know each other a little better.

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A subset of the artists at the celebratory post-exhibition pot-luck.  (Photo by Frank Farris.)

Finally, I wanted to mention that I am in the middle of my second time teaching Mathematics and Digital Art at the University of San Francisco.  I say this in the event you are interested in offering such a course at your university.  I have written extensively about this experience on my blog, and also have all course materials as well as a day-by-day outline available on the  Fall 2016 course website.  I would be happy to help you get such a course off the ground if you’re interested.

If you would like more information, or want to get in touch with any of the artists whose work is shown above, please make a comment and I’ll get back to you.  I hope this is just the beginning of a long tradition of having mathematical art exhibits at sectional MAA meetings!

On Coding VII: LaTeX I

I’ll talk about LaTeX in this installment of On Coding — and my next one, too.  There is so much to say, I realized I couldn’t say it all in just one blog post.

LaTeX is a markup language, like HTML, but its purpose is completely different.  TeX was invented when its creator, Donald Knuth, thought the galley proofs for one of his computer science textbooks looked so awful, he thought that something should be done about it.  This was in the late 70’s.  TeX was initially released in 1978; in 1985, Leslie Lamport released LaTeX, which is a more user-friendly version of TeX.  This Wikipedia article has a more complete history if you’re interested.

Basically, LaTeX makes math look great.  Here’s a formula taken from the paper Nick and I submitted to Bridges 2017 recently.

latex1

It’s perfectly formatted, all the symbols and spacing nicely balanced.  Without mentioning any names, try producing that same formula in some other well-known word-processing environment, and you’ll find out it doesn’t even come close to looking that good.

Before I go into more detail about all my favorite LaTeX features, I’d like to explain how it works.  Typically, you download some TeX GUI — I use TeXworks at home.  The environment looks like this:

latex2

On the left-hand side is the LaTeX markup, and on the right-hand side is a previewer which shows you how your compiled text would look as a pdf document.  Just type in your text, compile, view, repeat.  Much like HTML.

There are many things I like about LaTeX, and it’s hard to rank them in any particular order.  Although first — and foremost — mathematical formulas look fantastic.

A close second is the fact that it’s fast.  By that, I mean that because a markup language it text-based, there’s no mouse involved.  I’m a very fast typist, which means I can type LaTeX markup almost as fast as I can type ordinary text.  If you’ve ever hard to typeset a formula by means of drop-down menus, you know exactly what I mean.

The third feature is closely related to the second:  it’s intuitive.  For example, to get the trigonometric formula

\tan\dfrac{\theta}{2}=\dfrac{\sin\theta}{1+\cos\theta},

you would type

$$\tan\frac{\theta}{2}=\frac{\sin\theta}{1+\cos\theta}$$.

All commands in LaTeX are preceded by a backslash (“\”), so you can always distinguish them from text.  And if you look at the text, you can almost figure out the formula just from reading the commands.  It maps perfectly.

Most of LaTeX is that way — the commands describe what they do.  For example,

x\rightarrow y

is created using

$$x\rightarrow y$$.

Now you might be thinking that’s a lot to type for a simple formula — surely there must be a shorter way!  First, there is.  You can just define your own macro by saying

\def\ra{\rightarrow}

so you could just type

$$x\ra y$$.

This might be a good thing to do if you use a right arrow all the time.  But secondly, if you just use it occasionally, it’s really quite easy to remember.  When you want a right arrow, you just type “\rightarrow”.  If you want a longer arrow which points in both directions, like

x\longleftrightarrow y,

you just type

$$x\longleftrightarrow y$$.

Once you understand how the commands are named, it’s often easy to guess which one you’ll need just by thinking about it.

Next up, LaTeX makes your life a lot easier, especially when you’re working on big projects.  There are a lot of ways this is done, so I’ll just mention one of my favorites — the “\label” command.

\displaystyle\bigcup_{j\in{\mathbb Z}}{\bf R}^{2j}_{\theta}\,C_{r,\theta}^n={\bf R}^{n\,\rm{mod}\,2}_{\theta}\bigcup_{j\in{\mathbb Z}}{\bf R}^{2j}_{\theta}\,C_{1/r,\theta}^n\qquad(5)

This equation (again from the Bridges paper) is typeset using the commands (broken up for reference):

  1. \begin{equation}
  2. \bigcup_{j\in{\mathbb Z}}{\bf R}^{2j}_{\theta}\,C_{r,\theta}^n={\bf R}^{n\,\rm{mod}\,2}_{\theta}\bigcup_{j\in{\mathbb Z}}{\bf R}^{2j}_{\theta}\,C_{1/r,\theta}^n
  3. \label{theorem1}
  4. \end{equation}

The equation (described by (2)) is sandwiched between a begin/end block ((1) and (4)).  But the key command is the “\label” command on line (3).  When you want to refer to this equation in LaTeX, you don’t use text like “equation (5)”, you say “equation (\ref{theorem1})”.

The \label command assigns the number of the equation — in this case, 5 — to the label “theorem1”.  So when you use the “\ref” command (stands for “reference,” naturally), LaTeX will look for the number assigned to “theorem1”.

This might not seem like a big deal at first.  But as you work on a paper, you’re always deleting equations, adding them, or moving them around.  By assigning them labels, any references you make to equations in your text are automatically updated when you make changes.

And while we’re on the subject of equations — of course an extremely important topic when thinking about writing mathematics — there is also the “\nonumber” command, as well.  Before you end the equation, you migh add a \nonumber tag, as in

\begin{equation}<stuff>\label{eq1}\nonumber\end{equation}.

Why would you label an equation for easy reference, and then not even put the number next to the equation?  It is good mathematical style to only number equations that are referenced in the text.  If you just show them once and don’t refer to them later, they don’t need a number.

But as you rewrite a proof, for example, you might find you no longer need to reference a particular equation, and so you don’t need the number any more.  So rather than having to format it as not an equation (deleting the begin/end block), you just add the \nonumber tag.  It’s a lot easier.

So what I do is label every equation as I write, and then when I have a final draft, I just go through and unnumber all those equations which I never end up referencing.  It’s so nice.

I know I went on a bit about equations, but similar conveniences are available for figures, tables, article sections, book chapters, bibliographic entries, etc.  You never have to remember a number.  Ever.

And yes, there’s more….  Stay tuned for the next On Coding installment, where I’ll give you more reasons for wanting to learn LaTeX!

Mathematics and Digital Art: Update 1

I have the pleasure of teaching Mathematics and Digital Art again this semester!  Since I’m largely following my outline from last semester, biweekly reports aren’t really necessary.  But every month or so, I’d like to provide an update regarding changes I’ve made from the previous semester, as well as provide examples of student work.

There are no significant content changes yet — although I’ll be discussing L-systems rather than polyhedra this semester, and there will be more to say when we get to that point.  But as far as the delivery is concerned, there have been some alterations.

First, I’m emphasizing the code more right from the start.  You might recall that in their mid-semester comments last semester, students asked for more details about the actual coding.  So I take more time in each lab explaining Python.

This change has already made an impact; I’ve noticed that students are getting more adventurous with coding earlier on.  They really seem to enjoy experimenting with the geometry.  The example I use for the Josef Albers assignment looks like this — just rectangles within rectangles.

sample

But Collette took the geometry quite a few steps further.  In her narrative, she discussed working with figure and ground, trying to make each geometrically interesting.

Collette.png

I am pleased to see students playing so intently with the geometry.  At first, after a detailed discussion of using two-dimensional coordinates in Python, some students just tried randomly changing numbers to see what would happen.  But I encouraged them to be a little more intentional — that is, spend more time in the design stage — and they were largely successful.

The second change is that I spent an extra day on affine transformations at the beginning of our discussion, slowing down the pace a little.  Last semester, I recall that I needed to go back and review ideas I thought I covered in sufficient detail.  Hopefully, slowing down the pace will help.

In addition, I put together a summary of commonly used affine transformations, such as reflections:
guide

This seemed to be helpful — I used it for the linear algebra course I’m teaching as well, and students responded positively.  Feel free to look at it; just go to Day 6 on the course website.

The third change involves using discussion boards more deliberately on Canvas (which is our University’s content management system).  For each digital art assignment, I have students post drafts of their work, and have their peers comment on them.  Since I have a small class this semester (six students), it is not a problem to have each student comment on every other student’s work.

Students really seem to enjoy this, and I participate by writing comments as well.  But because everyone works at a different pace, some students lagged behind.  So now I’m being more formal about using the discussion board, and making it an assignment.

For example, the next assignment involves creating three pieces, and I have assigned students to upload drafts on Canvas by the beginning of class next Friday.  We’ll use Friday’s class so students can write and read comments; the assignment isn’t due until a few days later, so there will be time to incorporate new ideas into their drafts.

These changes are making a positive impact, and are making the course even more enjoyable this semester.  And I am also fortunate to have Nick Mendler as my course assistant again this semester, meaning there are two of us to work with students each day.  Students are really getting individual attention with their work.

Now let’s look at some more examples of student work!  For the assignment to create a color texture using randomness, Lainey worked to create an image which resembled a piece of fabric.

Lainey.png

For the Josef Albers assignment, Peyton (like Collette) also experimented a lot with the geometry of the individual elements.  She chose a color palette which reminded her of a succulent, and so created geometrical objects which represented spikes on a plant.

Peyton.png

And for the assignment based on my Evaporation piece, Karla chose a pink palette.  She looked at various values for the radius and the randomness in the radius so as to create a balance between overlapping circles and white space between the circles.

karla

Stay tuned for the next update!  In the next installment, I’ll let you know how the work with L-systems went.  One of my favorite topics…..