Geometrical Dissections III: Octagons and Dodecagons

It has been quite a while since I’ve written about geometrical dissections.  For a brief refresher, you might want to look at my first post on dissections.

But recently my interest has been rekindled — to the point that I started writing a paper a few weeks ago!  I often like to share mathematics I’m working on as it’s happening to give you some idea of the process of doing mathematics.  The paper has quite a bit more mathematics in it than I’ll include in this post, but I’ll try to give you the gist of what’s involved.

Recall (again, see my first post for a more thorough discussion) that I’m interested in finding dissections that you can draw on graph paper — I find these more enjoyable as well as being accessible to a wider audience.  So all the dissections I’ll talk about will be based on a grid.

The first example is a dissection of two octagons to one, such as shown below.


In other words, you can take the pieces from the two smaller octagons, move them around, and build the larger octagon.  If you look carefully, you’ll notice that the larger octagon is rotated 45° relative to the smaller octagons.  Geometrically, this is because to get an octagon twice the area of a given octagon, you scale the side lengths by \sqrt2.  Imagine a square — a square with side length S has area S^2, while a square with side length \sqrt2S has area (\sqrt2S)^2=2S^2  — which is double the area.

Now think of a segment of unit length.  To get a segment of length \sqrt2, you need to take a diagonal of a unit square, which rotates the segment by 45° as well as scales it by a factor of \sqrt2.  This is why the larger octagon is rotated with respect to the smaller ones.

But what makes this dissection interesting is that there is an infinite family of very similar dissections.  By slightly varying the parameters, you get  a dissection which looks like this:


Here, you can clearly see the 45° rotation of the larger octagon.  I always enjoy finding infinite families of dissections — it is very satisfying to discover that once you’ve found one dissection, you get infinitely many more for free!

The proof that this always works (which I will omit here!) involves using a geometrical argument — using arbitrary parameters — to show that the green triangles always have to be right isosceles triangles.  This is the essential feature of the dissection which makes it “work” all the time.

The second example I’m including in the paper is a dissection on a triangular grid, like the one below.


Note that the figure on the left is an irregular dodecagon; that is, it has twelve sides.  Recall that the interior angles of a regular dodecagon have measure 150° — but so do the angles of this irregular dodecagon as it is drawn on a triangular grid.

If you look carefully, you’ll see how the sides in the pieces of the dissection on the right match up with sides of the same length from the other pieces.  Also, looking at the dissection on the right, you’ll see that around all the interior vertices, there are two angles with measure 150° and one with measure 60°, adding up to 360° — as we would expect if the pieces fit exactly together.

And — as you might have guessed from the first example — this is also one of an infinite family of dissections.  As long as the sides of the irregular dodecagon alternate and the vertices stay on the triangular lattice, there is a dissection to a rhombus.  Below is another example, and there are infinitely many more….


My third and final example involves irregular dodecagons, but this time on a square grid.  And while I found the previous two dissections several years ago, I found this example just last week!  What inspired me was one of my favorite dissections — from an irregular dodecagon to a square — also found many years ago.


In the spirit of the first two examples, I asked myself if this dissection could be one of an infinite family.  The difficulty here was that there were three parameters determining an irregular dodecagon, as shown below.


We do need b>c so that the dodecagon is convex and actually has 12 sides; if b=c, four pairs of sides are in perfect alignment and the figure becomes an octagon.

There is an additional constraint, however, which complicates things a bit.  The area of this dodecagon must also be the area of some tilted square with vertices on the grid, as illustrated below.  Note that the area of this square is just d^2+e^2.


It is not a difficult geometry exercise to show that the area of the dodecagon is a^2+4(a+b)(c+b).  So in order to create a dissection, we must find a solution to the equation


Again, here is not the place to go into details.  But it is possible to find an infinite family of solutions when a=e.  You get a dissection which looks like this:


I was particularly pleased to have found this eight-piece dissection since most of my attempts until this point had ten pieces or more.  And to give you a sense of this family of dissections, here is an example with different parameters within this family.


You can definitely see the resemblance, but it is also clear that the dodecagon and square are not the same shape as those in the previous dissection.

So these are the three families of geometrical dissections I’ll be including in my paper.  I hope these examples might inspire you to pick up a pencil and graph paper and try to find some infinite families of your own!


Geometrical Dissections II: Four to One

This week, I’d like to discuss a piece of artwork which began as a geometrical dissection — I call it Four to One.


I thought it would be interesting to discuss the process of creating such a piece from beginning to end.  The creative process is not really mystical, but because we so often only see the finished product, it may seem that way at times.

It all began about 15 years ago, when I was teaching the Honors Geometry course I mentioned in last week’s post.  In Greg’s book Dissections Plane & Fancy, he takes a few chapters to discuss dissections from squares to other squares — frequently two different squares to one, or three to one.  But there was very little about four-to-one dissections, so I thought I’d explore this avenue a bit more.

I can’t recall precisely how I arrived at the identity


I might have written some for loops — but computers were not as fast back then….  Likely I used something like Lebesgue’s formula on p. 80 of Greg’s book, which gives a formula for creating three-to-one square dissections.  Then if one of those squares could be written as a sum of two others, I’d have a four-to-one dissection.  In particular, once I (might have!) found out that


I could use the fact that 15^2+36^2=39^2 (which is just a multiple of the Pythagorean triple (5, 12, 13)) to obtain the possibility of a four-to-one dissection described above.

Now this only suggested the puzzle, not the actual dissection itself.  And there certainly is a dissection — at the very least, we can cut up all the squares into 1\times1 unit squares and reassemble!

This is hardly an elegant solution; but I did come up with the following one:


I liked it because each square was cut into just two or three pieces, which was particularly nice.  Moreover, only one piece needed to be rotated.  But even though the number of pieces is relatively small, there is still the possibility that a dissection may exist using fewer pieces.

Of course my original solution was sketched on a yellowing piece of graph paper — but what to do with it now?

My first attempt looked like this:


I was thinking of creating various pathways through the dissected squares so that when they were rearranged, the pathways would still line up.  I abandoned this approach, however.  I can’t remember exactly why, but the results didn’t appeal to me — and besides, the paths themselves actually had nothing to do with the dissection puzzle itself.

But then I had the thought — which was in fact a real challenge — can I communicate what’s happening with the dissection using only one square?  In other words, could I depict the geometrical dissection by just showing the largest square without giving the viewer the four smaller squares?  I think what might have moved me in this direction is that there was just no elegant way of putting all five squares together in a composition.  There were just too many corners.

So I though of overlapping the smaller squares onto the largest square, as shown below (note:  you’ll notice an error in the geometry, but as it was a draft I discarded, I didn’t bother to fix it):



Now if you look very carefully, you can find all the pieces of the dissected squares in the largest square.  There is some overlap, of course — but smaller circles were overlaid on larger ones so colors from both circles could be seen.  (I copied the original dissection again so it’s easier to compare.  I used different colors as the images were created at different times, so watch out! )

I liked the idea — I felt I was getting somewhere.  But I wasn’t happy with the colors.  Now creating mathematical art makes you hungry — I can clearly recall driving to lunch while I was in the middle of this project, and I can even remember the road.  It was Fall in Princeton, NJ, and the leaves had already turned color.  No more oranges and reds — but lots of greens and yellows, as well as browns from the tree trunks.  My color palette!

What intrigued me about the idea was the fact that I was working with a very abstract, almost purely mathematical problem — and here I was, thinking about using organic colors from nature, from my life experience.

Now I had already been working with the ideas from Evaporation, and realized if I was using an organic palette, I couldn’t have the circles be regular, precise — and the colors couldn’t be pure either, just like you might find hundreds of shades of yellow in a Fall forest.


So, as shown in this close-up of Four to One, the colors were varied by using random numbers just as was done for Evaporation, but there were no extremes — each piece of each square had to be clearly recognizable if the dissection was to be clearly seen.

The sizes of the circles varied as well, helping to contribute to a natural texture.  Here, you can clearly see how smaller circles were overlaid on larger circles for the two-color effect.  The smaller circles, however, had only about one-fourth the area of the larger ones, so it was clear which color was dominant.

And there it is!  The creative process is not magical, not mystical — in fact, much of the time it seems to consist of failed inspiration….  Consider yourself lucky if your first attempt turns out to be your last as well — but more often than not, creativity is an iterative process involving constant revision.

So my advice is to stick to it!  Don’t worry if the first attempt isn’t what you imagine.  Now I used Mathematica to create this image — and I’ve been programming in Mathematica for over twenty years.  So I’m pretty good at taking an idea and implementing it fairly quickly.  But if you’re relatively new to programming, you’ve got to be patient with your programming skills as well.  I can tell you though — it’s worth it.  Don’t let anyone else tell you any different….

Geometrical Dissections I

Closely related to the problem of tiling the plane with polygons is that of dissecting one geometrical object into pieces that can be rearranged to form another. The classic example is the following dissection of an equilateral triangle to a square, attributed to Henry Dudeney, 1907.


Note that the pieces are exactly the same in both polygons. It’s not hard to appreciate the beauty and elegance of this “geometrical artwork.” And it’s not hard to imagine a puzzle based on this dissection — give the puzzler these four pieces with instructions to make both a triangle and a square from the same pieces.

This led me, once upon a time, to construct an Honors Geometry course centered around geometrical dissections using  Greg Frederickson’s wonderful Dissections Plane & Fancy.

But even “simple” dissections — involving triangles and squares — weren’t so easy to create. For example, the pieces divide the base of the equilateral triangle into lengthsDay014formulaand these are some of the easier calculations! I won’t say more about that here — but you can read all about this dissection and many others in Greg’s book.

So I was fascinated by geometrical dissections — but I needed a way to make the idea accessible to my students. I thought — what could you create just by experimenting with dissections on ordinary graph paper?

Well, I have been answering this question for over 15 years now. I’ll start with some introductory ideas in this post, but this is definitely not the last word on dissections!

Let’s begin with the following puzzle.  By cutting the rectangle along the grid lines, how many pieces are needed so you can also make the square with a corner missing?Day014Puzzle1AThis seems like an easy puzzle to solve, as shown below.Day014Puzzle1BSo yes, only two pieces are necessary — but one had to be rotated. Here is the question: can this puzzle be solved with just two pieces, but with neither piece rotated?

It turns out this is possible — but a solution requires a bit more creativity. Here is one way to do it:Day014Puzzle1CThis is a solution technique commonly used in Dissections Plane & Fancy.  Why bother?  In the world of geometric dissections (and it is a growing universe, surely, as any internet search will show), finding a minimum number of pieces is the primary objective.  But of all solutions with this minimum number of pieces, “nicer” solutions require rotating the fewest number of pieces.  And rotating none at all is — in an aesthetic sense — “best.”  It is also preferable not to turn pieces over, although sometimes this cannot be avoided for minimal solutions.

Another criterion for solutions might be whether they can be hinged or not, as Greg discusses extensively in Hinged Dissections:  Swinging & Twisting.  We won’t have time to explore this topic today.

Of course there is no reason you have to start with a rectangle, and also no reason why you need to restrict yourself to just one shape.  The puzzle below shows how you can find a four-piece, rotation-invariant dissection from two smaller octagons to one larger octagon.Day014Puzzle2It is important to note that the octagons here are not regular.  A quick glance through Dissections Plane & Fancy will reveal that dissections involving regular polygons are generally rather difficult (as the initial triangle-to-square example amply shows).

Further, “two-to-one” dissections lend themselves nicely to a square grid, as the diagonal of a unit square has length square root of 2.  Take a moment to study the octagon dissection again — paying particular attention to the side lengths — to see how this plays out.  In the world of regular polygons, however, two-to-one dissections are in general quite difficult.  Visit Gavin Theobald’s web page of two-to-one dissections to see some fascinating examples.

It is not hard to create dissection puzzles of your own — a pencil, eraser (!!!), and graph paper are all you need.  What is difficult, however, is proving that you’ve found the fewest number of pieces.  And when you have, proving that your dissection is unique.  Uniqueness is virtually impossible to prove, but sometimes you can get a handle on minimality.  For example, if the octagon dissection could be done in three pieces, one of the smaller octagons would have to be uncut.  It’s not hard to see that there is no way of cutting the other smaller octagon into just two pieces to create the larger octagon.

What I enjoy about creating dissection puzzles is that there is not a single strategy you can use to solve them.  You really need to use your imagination.  Sometimes you might even surprise yourself by stumbling upon a really neat puzzle, like the one below.

Here, an 8 x 8 square with four holes (shown in black) can be dissected into three pieces to create a 6 x 10 rectangle, although one piece needs to be rotated.  There is a simple elegance about this dissection which I find appealing.Day014Puzzle3

Another grid which lends itself to creating dissections is a grid of equilateral triangles.  We won’t go into details here, but you can get the idea with the following dissection of an irregular dodecagon to an equilateral triangle in just five pieces.  (The minimal dissection with regular polygons requires eight pieces.)Day014Puzzle4I’ll leave you with two puzzles to think about.  Of course, you can just make up your own.  If you come with anything interesting, feel free to comment!

For this puzzle, my best solution is four pieces, without needing to rotate any pieces or turn any pieces over.  (Black squares are holes.)Day014Puzzle5

For the last puzzle, my best solution is five pieces — but I had to turn over and rotate two of the pieces.Day014Puzzle6

A parting suggestion:  when looking up dissections on the web, be sure to use the search words “geometrical dissections…..”