truchet en plus

Since the previous post, I have been playing around with more variations on Truchet tiles (using this page). The variety of appealing patterns that you can create from these simple tiles is impressive.

the humble Truchet tile

For example, using this simple base tile you can create a path-like effect, even to the extent that paths can seem to weave under and over each other. The patterns below use this effect to suggest links and knots.

Truchet patters for two links (left)
and a trefoil knot (right)

Slight variations in the base tile can produce interesting effects. Here's an example that uses the traditional Truchet base tile.

Bulging the dark right triangle into a quarter circle allows us create something that looks quite different, even though it consists of exactly the same tile placements. Squares are replaced by circles, V-patterns replaced by tulips, and we end up with organic and densely packed patterns.

A common variation on the Truchet tile is the Smith tile, which consists of two quarter circles at opposite corners. Unlike the traditional Truchet, the Smith base tile has 180 degree symmetry, but because it lacks 90 degree symmetry, it can still be used to produce interesting and appealing patterns (if we had quarter circles in all four corners, the constructed patterns would always be the same - an array of circles). Using this tile, we end up with circles and blobby regions that create paths and zones across the grid.

A small change to the Smith tile allows us to eliminate the 180 degree symmetry and regain the expressiveness of the traditional tile patterns. For example replacing one of the quarter circles with a square means that some circles in our original pattern become squares, some remain circles, and some take on a lemon-shape, while the overall pattern retains the same topology as it has in the Smith version.

Another small change (using a diagonal line in place of the square) produces patterns that look quite different at fist: our paths now resemble strange jigsaw shapes. A closer look shows how the essential features are retained.

Even with all the possible variations, the original tile retains its charm. Can you see the four trefoils in the pattern below?

more pentagons, decagons, stars and rhombs

Kept adding on to one of the pentagon patterns from the previous post...

Eventually, ended up with  rough decagon shape - how many little decagons in there?

 

Zooming in, there are some nice patterns in there:

Yikes!

decorative fusings of the dodecagon

In the middle of playing around with various tiles and patterns, I lucked onto a post from Alex Bellos from back in February, in which he gives instructions for some decorative geometric constructions. He suggests dusting off a geometry set to implement them, but I found GSP worked nicely.

If you start with the line AB and follow Alex's instructions, you'll end up something like the above. Hiding what we want to hide, and mapping A and B onto the other corners of the square using an iteration will give you something that looks very close to his finished product.

In more decorative versions, the edges that extend out weave together in a braided fashion, producing an even more striking effect. But it was one of the later examples from the post that I was more interested in, as it seemed to show a more general method of using overlapping regular polygons (in this case, regular dodecagons) to produce decorative patterns. In one example, the dodecagons intersect at the midpoints of their sides at 90 degree angles:

I found that having the dodecagons intersect at the midpoints of their sides at 30 and 60 degree angles produces another interesting pattern.

If instead of intersecting at midpoints, we intersect the dodecagons at vertices, we can get other patterns. Here's one where the dodecagons have the smallest of overlaps.

Overlapping a little more, you get a pattern that you can make from a standard set of pattern blocks.

Still with a greater overlap, you can make a nice rosette.

And this pattern below has dodecagons overlapping in two ways (a bit hard to see the second):

Well, that was fun. Any other decorative patterns from overlapping dodecagons? I'll finish off with that dodecagon rosette pattern around a dodecagon.

regular polygons, in rings

Looking at the Kepler pentagonal tiling, you may notice the nice looking rings of pentagons around the decagons.

You can also make up other tilings with these rings of pentagons - to get the one below to work you have to sneak in some dented or overlapping pentagons.

But which regular n-gons can form rings like this? You obviously can't do it with a square.

And some regular n-gons, like heptagons, nonagons, decagons, and hendecagons (11-gons) don't work either.

All the angles of the regular n-gon are (n-2)pi/n - so the angles of the polygon in the center would have to be 4pi/n, but for that interior polygon to be a regular polygon itself, there must be some k for which the angle is also (k-2)pi/k. Equating these two values and solving for k gives k = 2n/(n-4). If we look for n that give integer values for k, then we have the n-gons that can form this sort of ring.

Which tells us that only the pentagon, hexagon, octagon, and dodecagon can form a ring around another regular n-gon (the regular decagon, hexagon, quadrilateral, and triangle, respectively). Coincidentally, these are the same polygons that can form a dented pinwheel, as described here.

But what if we skip over another edge (so 2 are skipped over) while forming the ring? We end up getting a star instead of a polygon in the center, and the smallest regular polygon this works for is the heptagon:

With a little bit of work, you may believe that this will work for n that give integer values for k = 2n/(n-6), and this turns out that those n values correspond to the regular heptagon, octagon, nonagon, decagon, dodecagon and octadecagon (18-gon).

As with the first kind of ring, the hendecagon fails:

But we can go further, and skip over another edge (3 now) when forming the ring of polygons. The center is no longer a star, but  a bumpy gear-like polygon, and the smallest regular polygon that can do this is the nonagon:

What other polygons can form this third kind of ring where 3 edges are skipped? Our function is now k = 2n/(n-8), and we get integer values for n = 9, 10, 12, 16, and 24.

Our hendecagons still won't form a ring when skipping 3 edges, but will once we start skipping 4.

If we skip m edges when putting the ring together, we can find the number of regularn-gons that will form the ring using the formula k = 2n/(n-2(m+1)), and will only get closed rings when k takes on integer values.

From this relationship we can find out a few things about these rings. For example, for any odd n, where n is 5 or more, we can form a ring by skipping (n-3)/2 edges and have a ring of 2n: for regular pentagons, we skip 1 edge and get a ring of 10, for heptagons we skip 2 edges and get a ring of 14, and for regular hendecagons, we skip 4 edges to get a ring of 22. Another observation: the eminently factorable 12 allows the dodecagon to form rings of 3, 4, 6, or 12.

I was lead to this while playing with regular heptagon, having fun making rings (and rings of rings) like the ones below.

regular polygons, intersecting regularly

Looking through the chapter on the number 5 in the really engaging book Single Digits: In Praise of Small Numbers, by Marc Chamberland, I came across an image and description of Kepler's pentagonal tiling, which looks like this:

Kepler Pentagonal Tiling

This tiling is made of pentagons, pentagrams, decagons, and fused decagons. Both the decagons and the fused decagons can be made from combinations of regular pentagons and dented pentagons (by dented, I mean in the way described here), so this tiling could also be made with pentagons, dented pentagons, and pentagrams.

Decagons and fused decagons from
pentagons and dented pentagons 

I wanted to try to make a similar tiling made only of the pentagons, dented pentagons, and their complementary rhombs, and found this:

Another pentagonal tiling

Which at first looks good - except these fused decagons are not the same as Kepler's - they have a greater overlap and cannot be made from pentagons and dented pentagons.

Encountering these two kinds of fused decagons, I wondered how many ways you can intersect two decagons, or other regular polygons, in the nice regular way these decagons were fusing.

Regular intersecting pentagons to nonagons

To overlap "regularly," the two n-gons must line up on their vertices. This means that the overlapping region will also be an equilateral polygon, that will have an even number of sides (half from each of the original n-gons), and will have all but two of its angles equal to (n-2)pi/n, as all but two of the angles will be angles from the original regular n-gons.

Since the overlapping region cannot have fewer sides/vertices than four, and because it can only be a polygon with an even number of sides/vertices, an n-gon will have [(n-4)/2] ways of regularly intersecting with itself (where the square bracket is the "ceiling" function, which tells you to round up any decimals). So, there are [(10-4)/2] = 3 of these nice fusings of the regular decagon.

The 3 fused decagons

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.