Friday, June 12, 2015

regular polygons, dented and sliced

A while ago, l noticed that sliced up octagons made nice tiles.
In particular, octagons that are split in a particular way into a dented octagon and a rhombus are pretty neat. These rhombuses are formed from so that they share with the octagon two adjacent sides of the octagon. The dented octagon is formed by slicing off the rhombus. Four of those dented octagons can be put around a vertex to form a pinwheel pattern, and four of the rhombs can be added to that pinwheel to make a bigger octagon.

You can do this sort of rhombic slicing with any regular polygon with more than 4 sides (you could slice a rhombus off a square in this way, if you consider the split off rhombus to be the same as the original square, but let's not go there). The picture below shows the angles you get when you slice a regular n-gon to get a dented n-gon and an n-rhomb.

One way to play around with these is to see how they can fit back together in various combinations. The motivating question is: For a given n, which combinations of regular n-gons, dented n-gons and n-rhombs can be placed around a vertex without gap or overlap?

Consider the pentagon
Lets start with a pentagon. The interior angles of the pentagon are 3pi/5. The small angle of the rhombus is going to be 2pi/5, and the small angle of the dented pentagon is going to be pi/5.

You may know that you can't place a set of non-overlapping pentagons around a vertex without leaving a gap (the best you can do is three, which give 9pi/5 around the vertex). Well, the dented pentagon you get from rhomb-splitting is just what you need to fill that gap (9pi/5 + pi/5 = 2pi).

The dented pentagons on their own can be placed around vertex nicely to form a decagon pinwheel, as can the rhombs, to form a star.

Rhombuses formed from "rhombic slicing" of a regular polygon can always be placed in a star pattern around a vertex. For an n-gon, the big angle of the sliced rhomb is always (n-2)pi/n, so the small angle is always 2pi/n, which means you can always place n of these small angles around a vertex without a gap or overlap.

You can't always place the dented polygons (small angle inward) formed by rhombic slicing around a vertex to form pinwheels like the dented pentagon, and you can't always use the dented polygons to fill in gaps formed by regular polygons around a vertex, but there are some combinations of all three (regular, dented, and rhomb) that will always fit around a vertex.

A few special arrangements
Suppose we wanted to put k regular polygons around a vertex. The sum of the angles around the vertex would have to be 2pi. So we would have k(n-2)pi/n = 2pi. Solving for k, we have k = 2n/(n-2). This only has 3 integer solutions, which occur when n = 3, 4, and 6. So from this we know we can arrange 6 equilateral triangles, 4 squares, or 3 regular hexagons around a vertex, but no other single regular polygon.

If we want to put k dented polygons around a vertex to make a pinwheel like we saw above for the pentagons, we'd place the small angles at the vertex and try to have them sum to 2pi. This requires us to have k(n-4)pi/n = 2pi, or  k = 2n/(n-4). So only n = 5, 6, 8, or 12 will work.

It was neat how the dented pentagon could be used to fill the gap left by an arrangement of regular pentagons; can we do that for any other regular polygons? It turns out that octagons are the only other regular polygon that fits together with one of its "dented" selves in this way. Here we are looking at k(n-2)pi/n + (n-4)pi/n = 2pi, which gives us k = (n+4)/(n-2), and k will be an integer only for n = 5 and n = 8.

Above we saw that regular polygons with 5, 6, 8, and 12 sides would, when dented, form pinwheels. There may be some cases where an incomplete pinwheel can be completed with a full regular polygon (sort of the opposite case of what we just saw with a dented polygon completing an arrangement of regular polygons). For this we are looking at (n-2)pi/n + k(n-4)pi/n = 2pi, which gives us k = (n+2)/(n-4), with integer values of k at n = 5, 6, 7, and 10.

If we throw the sliced rhombuses into the mix, we get more options for arranging tiles around a vertex. For example, when can you arrange one regular polygon with k of its sliced rhombuses around a vertex? Here the equation is  (n-2)pi/n + 2kpi/n = 2pi, or k = (n+2)/2, which gives us integer k values for all even n.  The resulting squid-like shapes alternate between having an even number or odd number of limbs.

What always works
We already noticed that you can always place n of the rhombs from an n-gon around a vertex without a gap to form a star. There are some other combinations of these tiles that also always work.

For example, 2 regular polygons and 2 corresponding rhombs will work. 2(n-2)pi/n + 2(2pi)/n = 2pi. But then, you can take one of those two regular polygons and split it into a dented polygon and a rhomb, so 1 regular polygon, 1 dented polygon and 3 rhombs will work. Finally you can split your last regular polygon to get a combination of 2 dented polygons and 4 rhombs.

Saturday, May 30, 2015

octo rhomb

Regular octagons cannot be used to tile by themselves - if you try, you will find there are square gaps that need to be filled.

If you slice a rhombus off your octagon, you'll end up with two tiles - a rhombus and a dented octagon.
Each of these shapes can be used to tile by themselves, or tile together. The rhomb-by-itself tiling is easy to visualize (imagine a squashed grid), here is the dented-octagon tiling:

Now, here's something else: you can take four of these rhombically challenged octagons to make a bigger octagon:

You get a nice tree shape if you remove two rhombs from the original octagon.

And these two shapes can be combined in many different ways.

If you remove 3 rhombs from the octagon, you get a shape that's a combination of two squares and another rhomb - a bent spiky thing:

Tuesday, April 14, 2015

octagonal iteration with GSP

Here is a little GSP iteration that I came across that I thought was worth sharing.

Start with a line segment - this provides the only "free" points in the sketch - everything else is constructed on top of this, starting with a square based on AB.

Next, construct the center of the square, and a circle centered on that square's center and diameter equal to the diagonal of the square.

Next, construct points on the circle midway between the points provided by the corners of the squares.

We'll iterate by mapping the original line segment onto pairs of these points (in GSP you can select the free points and map them onto other points derived from them). Mapping AB onto the pairs of adjacent points around the circle starts out like this:

And continuing on around the circle:

And so on ...

Until you get this:

Which is nice, but to unpack it a bit and see more of the patterns you can reduce the number of iterations and hide the original construction. Scaling back to one iteration, and hiding everything except the squares around the circle, you get this:

Simplifying even further you can just leave one side of the generated squares, say one of the sides that radiate out from the octagon in the middle.

If you start with this, the next two generations look something like stick men chasing each other, 

If instead you hide everything but the outermost edges of the squares around the central hexagon, you would begin with this exploded octagon:

And with two more generations you would have this pleasing pattern (finding the overlapping octagons below is not easy):

My favorite in this series is what you get from starting with the innermost edges of the squares - the octagon itself.

As you add more generations to this, the overlapping octagons slice each other into an interesting tessellation made up of octagons, squares, hexagons, triangles and a few other tiles that can be thought of as slices of octagons or shapes that fill in the gaps between them.

Here are some of the tiles suggested by this pattern:

Adding another generation produces even more tiles, formed by slicing and intersecting the tiles from the previous:

What if you started this whole thing with a triangle instead of a square? You might end up with a tessellation like this:

Wednesday, March 25, 2015

are you experienced?

Don't despair
A search for "math" in the iTunes store is likely to disappoint (maybe "maths" or "mathematics" would provide better results). I haven't tried Math Drills Lite - it is likely the last thing I would want to download, yet it comes up first.

A sad situation

But this is happy post, because there is a math app, well, more of an interactive book, that is engaging, interesting, well written, and attractively designed, that conveys mathematics as its practitioners and enthusiasts see it: beautiful and creative, not dry and confusing. Mathema, written by two mathematicians, Hugo Parlier and Paul Turner, is an accessible math app/book for students, teachers, and general folk that seeks to provide its readers (interactors?) with authentic mathematical experiences that capture the process and feelings associated with doing mathematics.

What does doing mathematics feel like?
Structured around three core mathematical experiences, Mathema presents visually interesting puzzles and games, and then proceeds to introduce the math that can be used to make sense of them. Along the way, we get excursions into graph theory, metric spaces, algebraic structures, and other advanced topics (for a popular book), but always grounded in answering questions that naturally arise in each investigation.

The Dominoes Puzzle: its solvabilty 
depends on where you put the holes

The first mathematical experience includes some familiar mathematical recreations made fresh by the interactive capabilities of the "book." The process of doing mathematics (encountering a problem, making a conjecture, using logical reasoning, and benefiting from some key insight) and the feelings of doing math (enjoyment, frustration, satisfaction) are presented through the "Dominoes Puzzle": can you cover a chessboard with two missing squares with dominoes without overlap or gaps? I shared this part of the experience with a ten year old, who enjoyed trying to cover the board missing corners with the dominoes, agreed with the hypothesis that it was impossible after trying it a few times, and then listened to and understood the proof that explained why it could not be done. I'd suggest that the target audience for the rest of the book is a bit older, but it was very nice to see how well and simply this first part of the book accomplished its goal.

Mathematical Games for a new generation
The Dominoes Puzzle might be familiar to some as the "mutilated chessboard" problem from  Martin Gardner's first Scientific American collection (problem 3 in Nine Problems), but the presentation in Mathema is much more complete and inviting. Actually being able to place the dominoes on the virtual (unmarked) chessboard is a great advantage of the ebook format (no chessboards were harmed, no actual dominoes required), and being able to move the holes around extends the puzzle meaningfully. The first experience also includes an explanation of why the game Hex (also written about by Gardner in his first book of Scientific American columns, who tells us that the game was co-invented by Piet Hein and John Nash) can always be won. Here again, the interactive nature of the book is used to advantage. Back in the 1950s, Martin Gardner suggested to readers that a Hex board of their own "can easily be drawn on heavy cardboard or made by cementing together hexagonal tiles." The software version of Hex in Mathema beats cardboard and cement, and being able to quickly generate many complete honeycomb patterns leads naturally to the hypothesis that you will always have a path from one side of the board to its opposite.

A portion of a Honeycomb pattern:
these tell us about the winnability of Hex

The "chroma square" puzzle of the second experience is suggestive of a two dimensional Rubik's cube, and learning the trick to solving them quickly gave me an unreasonably great sense of accomplishment. The idea of treating these puzzles as mathematical objects and considering the space of all chroma squares is well presented, and the excursion into abstraction is repaid when the method for solving the puzzle is used to prove a couple of interesting things about all possible chroma squares and how they relate to each other. The third and final mathematical experience allows you to play with a more dynamic system of "flows." At first flows seem to bear little resemblance to the puzzles of the first two sections, but Mathema shows that by a thoughtful process of making definitions, these too can be analyzed using mathematical thinking.

Fans blow particles within a disk
creating a flow

Why does it work?
The visual and interactive way Mathema is designed, with its scrolling, flipping, and zooming, is a significant part of its appeal and its ability to engage. I think there are more important things that it does right that have nothing to do with technology, however. How the authors understand authentic learning experiences is key. What is "authentic" math? - the mistake that some people continue make is to assume that to make mathematics interesting and relevant it must be connected to some practical or real-world application. Puzzles, patterns, games, and aesthetically interesting images are the real hooks that make math interesting, and understanding this is part of Mathema's appeal. Another thing Mathema does right is recognizing that mathematics is most relevant and interesting when it is presented in a non-curricular way: not as topics that are isolated from each other and blocked off from amateurs, but instead as a way of making sense of the world that is available to anyone who wants to use it, and applicable almost anywhere.

Thursday, March 19, 2015

a tile arrangement, or airport fun

Is there anything nicer than a notebook with grid-lined pages? Maybe, but they are pretty nice - and I count myself very fortunate to have just obtained a new one. 

And thanks mostly to a longish wait in the Vancouver airport, this is what ended up on page one.

The image on the tiles are the simplest non-trivial knot, the trefoil, which you could also put together using these other tiles.

Along a given row or column (following the slight skew), the tiles are alternately rotated back and forth by 90 degrees - in the rows they alternate between being placed at 0 and 90 degrees or at 270 and 180 degrees, and down the columns they alternate between being placed at 0 and 270 degrees or at 90 and 180 degrees. This placement allows the tiles to rotate as they revolve around each of the smaller black squares (which are 1/16th the size of the trefoil tile). In the picture below, as you follow the tiles clockwise around the small squares marked A the trefoil tiles rotate clockwise, but as you follow the tiles clockwise around B they rotate counterclockwise.