Polygons

In working on a proposal for a book about three-dimensional polyhedra last week, I needed to write a brief section on polygons.  I found that there were so many different types of polygons with such interesting properties, I thought it worthwhile to spend a day talking about them.  If you’ve never thought a lot about polygons before, you might be surprised how much there is to say about them….

So start by imagining a polygon — make it a pentagon, to be specific.  Try to imagine as many different types of pentagons as you can.

How many did you come up with?  I stopped counting when I reached 20….

Likely one of the first to come to mind was the regular pentagon — five equal sides at angles of 108° from each other.  Question:  did you only think of the vertices and edges, or did you include the interior as well?

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Why consider this question?  An important geometrical concept is that of convexity.  A convex polygon property has the property that a line segment joining any two points in the polygon lies entirely within the polygon.

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The two polygons on the left are convex, while the two on the right are not.  But note that for this definition to make any sense at all, a polygon must include all of its interior points.

Convex polygons have various properties.  For example, if you take the vertices of a convex polygon and imagine stretching a rubber band beyond the vertices and letting it snap back, the rubber band will describe the edges of the polygon.  See this Wikipedia article on convex polygons for more properties of convex polygons.

Did the edges of any of your pentagons cross each other, like the one on the left below?

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In this picture, we indicate vertices with dots to illustrate that this is in fact a pentagon.  The points where the edges cross are not considered vertices of the polygon.  The polygon on the right is actually a nonconvex decagon, even though it bears a resemblance to the pentagon on the left.

But not so fast!  If you ask Mathematica to draw the polygon on the left with the five vertices in the order they are traversed when drawing the edges, here is what you get:

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So what’s going on here?  Why is the pentagon empty in the middle?  When I gave the same instructions using Tikz in LaTeX (which is how I created the light blue pentagram shown above), the middle pentagon was filled in.

Some computer graphics programs use the even-odd rule when drawing self-intersecting polygons.  This may be thought of in a few ways.  First, if you imagine drawing a segment from a point in the interior pentagon to a point outside, you have to cross two edges of the pentagon, as shown above.  If you draw a segment from a point in one of the light red regions to a point outside, you only need to cross one edge.  Points which require crossing an even number of edges are not considered as being interior to the polygon.

Said another way, if you imagine drawing the pentagram, you will notice that you are actually going around the interior pentagon twice.  Any region traversed twice (or an even number of times) is not considered interior to the polygon.

Why would you want to color a polygon in this way?  There are mathematical reasons, but if you watch this video by Vi Hart all the way through, you’ll see some compelling visual evidence why you might want to do this.

We call polygons whose edges intersect each other self-intersecting or crossed polygons.  And as you’ve seen, including the interiors can be done in one of two different ways.

But wait!  What about this polygon?  Can you really have a polygon where a vertex actually lies on one of the edges?

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Again, it all depends on the context.  I think you’re beginning to see that the question “What is a pentagon?” is actually a subtle question.  There are many features a pentagon might have which you likely would not have encountered in a typical high school geometry course, but which still merit some thought.

Up to now, we’ve just considered a polygon as a two-dimensional geometrical object.  What changes when you jump up to three dimensions?

Again, it all depends on your definition.  You might insist that a polygon must lie in a plane, but….

It is possible to specify a polygon by a list of points in three dimensions — just connect the points one by one, and you’ve got a polygon!  Of course with this definition, many things are possible — maybe you can repeat points, and maybe the points do not all lie in the same plane.

An interesting example of such a polygon is shown below, outlined in black.

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It is called a Petrie polygon after the mathematician who first described it.  In this case, it is a hexagon — think of holding a cube by two opposite corners, and form a hexagon by the six edges which your fingers are not touching.

There is a Petrie polygon for every Platonic solid, and may be defined as follows:  it is a closed path of connected edges such that no three consecutive edges belong to the same face.  If you look at the figure above, you’ll find this is an alternative way to define a Petrie hexagon on a cube.

And if that isn’t enough, it is possible to define a polygon with an infinite number of sides!  Just imagine the following jagged segment continuing infinitely in both directions.

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This is called an apeirogon, and may be used to study the tiling of the plane by squares, four meeting at each vertex of the tiling.

And we haven’t even begun to look at polygons in other geometries — spherical geometry, projective geometry, inversive geometry….

Suffice it to say that the world of polygons is much more than just doodling a few triangles, squares or pentagons.  It is always amazes me how such a simple idea — polygon — can be the source of such seemingly endless investigation!  And serve as another illustration of the seemingly infinite diversity within the universe of Geometry….

Bridges 2017 in Waterloo, Canada!

Bridges 2017 was in full swing last weekend, so now it’s time to share some of the highlights of the conference.  Seems like they keep getting better each year!

The artwork was, as usual, quite spectacular.  I’ll share a few favorites here, but you can go to the Bridges 2017 Gallery to see all the pieces in the exhibitions, along with descriptions by the artists.

My favorite painting was Prime numbers and cylinders by Stephen Campbell.

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What is even more amazing than the piece itself is that Stephen makes his own paint!  So each piece involves an incredible amount of work – and the results are worth it.  Visit Stephen’s website to see more of his work and learn a bit more about his artistic process.

I also liked these open tilings of space by Frank Gould.

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Although they are simple in design, the overall effect is quite appealing.  I often find that the fewer elements in a piece, the more difficult it is to coax them into interacting in a meaningful way.  It is challenging to be a minimalist.

This lantern, Variations on Colourwave 17 — Mod 2, designed by Eva Knoll, shows a pattern created using modular arithmetic.

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The plenary talks this year were almost certainly the best I have ever experienced at any conference.  Opening the conference was a talk by Damian Kulash of the band OK Go.  He described his creative process and gave some insight to how he makes his unbelievable videos.

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They really cannot be described in words – one example he talked about was this video filmed in zero gravity in an airplane!  There are no tricks here — just unbridled creativity and cleverness.

Another favorite was the talk by John Edmark.  Many of his ideas had a spiral theme, like the piece you see below.

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But what is fascinating about his work is how it moves.  When this structure is folded inward, it actually stretches out.  This is difficult to describe in words, but you can see a video of this phenomenon on John’s website.  What made his talk really interesting is that he discussed the mathematics behind the design of his work.

Stephen Orlando talked about his motion exposure photography.

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Although seemingly impossible, this image is just one long-exposure photograph — the colors you see were not added later.  Stephen’s technique is to put programmable LED lights on a paddle and photograph a kayaker paddling across the water.

But where is the kayaker?  Notice the background — it’s almost dark.  It turns out that if the kayaker moves at a fast enough rate, the darkness does not allow enough time for an image of the kayaker to be captured.  Simply amazing.  Visit Stephen’s website for more examples of his motion exposure photographs.

There were also many other interesting talks given by Bridges participants, but there is not enough room to talk about them all.  You can always go to the Bridges 2017 website and download any of the papers you’re interested in reading.

But Bridges is also more than just art and talks.  Bridges participants are truly a community of like-minded people, so social and cultural events are also an important aspect of any Bridges conference.

I shared an AirBnB with Nick and his parents a short walk from campus — the house was spacious and comfortable, and really enhanced the Bridges experience.  Sandy (Nick’s mother) wanted to host a gathering, so I invited several friends and colleagues over the Friday night of the conference for an informal get-together.

It was truly an inspiring evening!  There were about fifteen of us, mostly from around the Bay area.  Everyone was talking about mathematics and art — we were so engrossed, it turns out that no one even remembered to take any pictures!  One recurring topic of discussion was the possibility of having some informal gatherings throughout the year where we could share our current thoughts and ideas.  I think it may be possible to use space at the University of San Francisco — I’ve already begun looking into it.

Lunch each day was always from 12:00–2:00, so there was never any need to rush back.  This left plenty of time for conversation, and often allowed time to admire the art exhibitions.  It seems that you always noticed something new every time you walked through the displays.

On Saturday evening of the conference was a choir concert featuring a cappella voices singing a wide range of pieces spanning from the 15th century to the present day.  It was a very enjoyable performance; you can read more about it here.

The evening of Sunday, July 30th, was the last evening of the conference for us.  Many participants were going to Niagara Falls the next day, but we were all flying out on Monday.  We decided to find a group and go out for dinner — and about a dozen of us ended up at a wonderful place for pizza called Famoso.

We chatted for quite some time, and then split up — some wanted to attend an informal music night/talent show on campus, but others (including me and the Mendlers) went to shoot some pool at a local pool hall.  Again, a good time was had by all.

While the first day of the conference seemed a little slow, the rest just flew by.  Another successful Bridges conference!  Nick and I both had artwork exhibited and gave talks, caught up with friends and colleagues from all around the globe, enjoyed many good meals, and got our fill of mathematical art.

Although the location for the next Bridges conference is usually announced at the end of the last plenary talk, these is still some uncertainty about who will be hosting the conference next year.  But regardless of where it is, you can expect that Nick and I will certainly be there!