Bent Pent

This is a beautiful and interesting shape, reminiscent of a piece of jewellery:

Pentagons in a ring


I came across it in this tricky little word-puzzle:

Word puzzle using pentagon-ring


Here’s a printable version of the puzzle:

Printable puzzle


Let’s try placing some other regular polygons with s sides around regular polygons with s*2 sides:

Hexagonal ring of triangles


Octagonal ring of squares


Decagonal ring of pentagons


Dodecagonal ring of hexagons


Only regular pentagons fit perfectly, edge-to-edge, around a regular decagon. But all these polygonal-rings can be used to create interesting and beautiful fractals, as I hope to show in a future post.

This Charming Dis-Arming

One of the charms of living in an old town or city is finding new routes to familiar places. It’s also one of the charms of maths. Suppose a three-armed star sprouts three half-sized arms from the end of each of its three arms. And then sprouts three quarter-sized arms from the end of each of its nine new arms. And so on. This is what happens:

Three-armed star


3-Star sprouts more arms


Sprouting 3-Star #3


Sprouting 3-Star #4


Sprouting 3-Star #5


Sprouting 3-Star #6


Sprouting 3-Star #7


Sprouting 3-Star #8


Sprouting 3-Star #9


Sprouting 3-Star #10


Sprouting 3-Star #11 — the Sierpiński triangle


Sprouting 3-star (animated)


The final stage is a famous fractal called the Sierpiński triangle — the sprouting 3-star is a new route to a familiar place. But what happens when you trying sprouting a four-armed star in the same way? This does:

Four-armed star #1


Sprouting 4-Star #2


Sprouting 4-Star #3


Sprouting 4-Star #4


Sprouting 4-Star #5


Sprouting 4-Star #6


Sprouting 4-Star #7


Sprouting 4-Star #8


Sprouting 4-Star #9


Sprouting 4-Star #10


Sprouting 4-star (animated)


There’s no obvious fractal with a sprouting 4-star. Not unless you dis-arm the 4-star in some way. For example, you can ban any new arm sprouting in the same direction as the previous arm:

Dis-armed 4-star (+0) #1


Dis-armed 4-Star (+0) #2


Dis-armed 4-Star (+0) #3


Dis-armed 4-Star (+0) #4


Dis-armed 4-Star (+0) #5


Dis-armed 4-Star (+0) #6


Dis-armed 4-Star (+0) #7


Dis-armed 4-Star (+0) #8


Dis-armed 4-Star (+0) #9


Dis-armed 4-Star (+0) #10


Dis-armed 4-star (+0) (animated)


Once again, it’s a new route to a familiar place (for keyly committed core components of the Overlord-of-the-Über-Feral community, anyway). Now try banning an arm sprouting one place clockwise of the previous arm:

Dis-armed 4-Star (+1) #1


Dis-armed 4-Star (+1) #2


Dis-armed 4-Star (+1) #3


Dis-armed 4-Star (+1) #4


Dis-armed 4-Star (+1) #5


Dis-armed 4-Star (+1) #6


Dis-armed 4-Star (+1) #7


Dis-armed 4-Star (+1) #8


Dis-armed 4-Star (+1) #9


Dis-armed 4-Star (+1) #10


Dis-armed 4-Star (+1) (animated)


Again it’s a new route to a familiar place. Now trying banning an arm sprouting two places clockwise (or anti-clockwise) of the previous arm:

Dis-armed 4-Star (+2) #1


Dis-armed 4-Star (+2) #2


Dis-armed 4-Star (+2) #3


Dis-armed 4-Star (+2) #4


Dis-armed 4-Star (+2) #5


Dis-armed 4-Star (+2) #6


Dis-armed 4-Star (+2) #7


Dis-armed 4-Star (+2) #8


Dis-armed 4-Star (+2) #9


Dis-armed 4-Star (+2) #10


Dis-armed 4-Star (+2) (animated)


Once again it’s a new route to a familiar place. And what happens if you ban an arm sprouting three places clockwise (or one place anti-clockwise) of the previous arm? You get a mirror image of the Dis-armed 4-Star (+1):

Dis-armed 4-Star (+3)


Here’s the Dis-armed 4-Star (+1) for comparison:

Dis-armed 4-Star (+1)


Elsewhere other-accessible

Boole(b)an #2 — other routes to the fractals seen above

Trifylfots

Here’s a simple fractal created by dividing an equilateral triangle into smaller equilateral triangles, then discarding (and rotating) some of those sub-triangles, then doing the same to the sub-triangles:

Fractangle (triangle-fractal) (stage 1)


Fractangle #2


Fractangle #3


Fractangle #4


Fractangle #5


Fractangle #6


Fractangle #7


Fractangle #8


Fractangle #9


Fractangle (animated)


I’ve used the same fractangle to create this shape, which is variously known as a swastika (from Sanskrit svasti, “good luck, well-being”), a gammadion (four Greek Γs arranged in a circle) or a fylfot (from the shape being used to “fill the foot” of a stained glass window in Christian churches):

Trifylfot


Because it’s a fylfot created ultimately from a triangle, I’m calling it a trifylfot (TRIFF-ill-fot). Here’s how you make it:

Trifylfot (stage 1)


Trifylfot #2


Trifylfot #3


Trifylfot #4


Trifylfot #5


Trifylfot #6


Trifylfot #7


Trifylfot #8


Trifylfot #9


Trifylfot (animated)


And here are more trifylfots created from various forms of fractangle:













































Elsewhere other-accessible

Fractangular Frolics — more on fractals from triangles

Circus Trix

Here’s a trix, or triangle divided into six smaller triangles:

Trix, or triangle divided into six smaller triangles


Now each sub-triangle becomes a trix in its turn:

Trix stage #2


And again:

Trix #3


Trix #4


Trix #5


Trix divisions (animated)


Now try dividing the trix and discarding sub-triangles, then repeating the process. A fractal appears:

Trix fractal #1


Trix fractal #2


Trix fractal #3


Trix fractal #4


Trix fractal #5


Trix fractal #6


Trix fractal #7


Trix fractal (animated)


But what happens if you delay the discarding, first dividing the trix completely into sub-triangles, then dividing completely again? You get a more attractive and symmetrical fractal, like this:

Trix fractal (delayed discard)


And it’s easy to convert the triangle into a circle, creating a fractal like this:

Delayed-discard trix fractal converted into circle


Delayed-discard trix fractal to circular fractal (animated)


Now a trix fractal that looks like a hawk-god:

Trix hawk-god #1


Trix hawk-god #2


Trix hawk-god #3


Trix hawk-god #4


Trix hawk-god #5


Trix hawk-god #6


Trix hawk-god #7


Trix hawk-god (animated)


Trix hawk-god converted to circle


Trix hawk-god to circle (animated)


If you delay the discard, you get this:

Trix hawk-god circle (delayed discard)


And here are more delayed-discard trix fractals:







Various circular trix-fractals (animated)


Post-Performative Post-Scriptum

In Latin, circus means “ring, circle” — the English word “circle” is actually from the Latin diminutive circulus, meaning “little circle”.

Fractangular Frolics

Here’s an interesting shape that looks like a distorted and dissected capital S:

A distorted and dissected capital S


If you look at it more closely, you can see that it’s a fractal, a shape that contains itself over and over on smaller and smaller scales. First of all, it can be divided completely into three copies of itself (each corresponding to a line of the fractangle seed, as shown below):

The shape contains three smaller versions of itself


The blue sub-fractal is slightly larger than the other two (1.154700538379251…x larger, to be more exact, or √(4/3)x to be exactly exact). And because each sub-fractal can be divided into three sub-sub-fractals, the shape contains smaller and smaller copies of itself:

Five more sub-fractals


But how do you create the shape? You start by selecting three lines from this divided equilateral triangle:

A divided equilateral triangle


These are the three lines you need to create the shape:

Fractangle seed (the three lines correspond to the three sub-fractals seen above)


Now replace each line with a half-sized set of the same three lines:

Fractangle stage #2


And do that again:

Fractangle stage #3


And again:

Fractangle stage #4


And carry on doing it as you create what I call a fractangle, i.e. a fractal derived from a triangle:

Fractangle stage #5


Fractangle stage #6


Fractangle stage #7


Fractangle stage #8


Fractangle stage #9


Fractangle stage #10


Fractangle stage #11


Here’s an animation of the process:

Creating the fractangle (animated)


And here are more fractangles created in a similar way from three lines of the divided equilateral triangle:

Fractangle #2


Fractangle #2 (anim)

(open in new window if distorted)


Fractangle #2 (seed)


Fractangle #3


Fractangle #3 (anim)


Fractangle #3 (seed)


Fractangle #4


Fractangle #4 (anim)


Fractangle #4 (seed)


You can also use a right triangle to create fractangles:

Divided right triangle for fractangles


Here are some fractangles created from three lines chosen of the divided right triangle:

Fractangle #5


Fractangle #5 (anim)


Fractangle #5 (seed)


Fractangle #6


Fractangle #6 (anim)


Fractangle #6 (seed)


Fractangle #7


Fractangle #7 (anim)


Fractangle #7 (seed)


Fractangle #8


Fractangle #8 (anim)


Fractangle #8 (seed)


Hour Re-Powered

Pre-previously on Overlord in terms of the Über-Feral, I looked at my favorite member of the fractal community, the Hourglass Fractal:

The hourglass fractal


A real hourglass for comparison


As I described how I discovered the hourglass fractal indirectly and by accident, then showed how to create it directly, using two isosceles triangles set apex-to-apex in the form of an hourglass:

Triangles to hourglass #1


Triangles to hourglass #2


Triangles to hourglass #3


Triangles to hourglass #4


Triangles to hourglass #5


Triangles to hourglass #6

[…]

Triangles to hourglass #10


Triangles to hourglass #11


Triangles to hourglass #12


Triangles to hourglass (animated)


Now, here’s an even simpler way to create the hourglass fractal, starting with a single vertical line:

Line to hourglass #1


Line to hourglass #2


Line to hourglass #3


Line to hourglass #4


Line to hourglass #5


Line to hourglass #6


Line to hourglass #7


Line to hourglass #8


Line to hourglass #9


Line to hourglass #10


Line to hourglass #11


Line to hourglass (animated)


Hour Power

Would it be my favorite fractal if I hadn’t discovered it for myself? It might be, because I think it combines great simplicity with great beauty. I first came across it when I was looking at this rep-tile, that is, a shape that can be divided into smaller copies of itself:

Rep-4 L-Tromino


It’s called a L-tromino and is a rep-4 rep-tile, because it can be divided into four copies of itself. If you divide the L-tromino into four sub-copies and discard one particular sub-copy, then repeat again and again, you’ll get this fractal:

Tromino fractal #1


Tromino fractal #2


Tromino fractal #3


Tromino fractal #4


Tromino fractal #5


Tromino fractal #6


Tromino fractal #7


Tromino fractal #8


Tromino fractal #9


Tromino fractal #10


Tromino fractal #11


Hourglass fractal (animated)


I call it an hourglass fractal, because it reminds me of an hourglass:

A real hourglass


The hourglass fractal for comparison


I next came across the hourglass fractal when applying the same divide-and-discard process to a rep-4 square. The first fractal that appears is the Sierpiński triangle:

Square to Sierpiński triangle #1


Square to Sierpiński triangle #2


Square to Sierpiński triangle #3


[…]


Square to Sierpiński triangle #10


Square to Sierpiński triangle (animated)


However, you can rotate the sub-squares in various ways to create new fractals. Et voilà, the hourglass fractal appears again:

Square to hourglass #1


Square to hourglass #2


Square to hourglass #3


Square to hourglass #4


Square to hourglass #5


Square to hourglass #6


Square to hourglass #7


Square to hourglass #8


Square to hourglass #9


Square to hourglass #10


Square to hourglass #11


Square to hourglass (animated)


Finally, I was looking at variants of the so-called chaos game. In the standard chaos game, a point jumps half-way towards the randomly chosen vertices of a square or other polygon. In this variant of the game, I’ve added jump-towards-able mid-points to the sides of the square and restricted the point’s jumps: it can only jump towards the points that are first-nearest, seventh-nearest and eighth-nearest. And again the hourglass fractal appears:

Chaos game to hourglass #1


Chaos game to hourglass #2


Chaos game to hourglass #3


Chaos game to hourglass #4


Chaos game to hourglass #5


Chaos game to hourglass #6


Chaos game to hourglass (animated)


But what if you want to create the hourglass fractal directly? You can do it like this, using two isosceles triangles set apex-to-apex in the form of an hourglass:

Triangles to hourglass #1


Triangles to hourglass #2


Triangles to hourglass #3


Triangles to hourglass #4


Triangles to hourglass #5


Triangles to hourglass #6


Triangles to hourglass #7


Triangles to hourglass #8


Triangles to hourglass #9


Triangles to hourglass #10


Triangles to hourglass #11


Triangles to hourglass #12


Triangles to hourglass (animated)


Fractal + Star = Fractar

Here’s a three-armed star made with three lines radiating at intervals of 120°:

Triangular fractal stage #1


At the end of each of the three lines, add three more lines at half the length:

Triangular fractal #2


And continue like this:

Triangular fractal #3


Triangular fractal #4


Triangular fractal #5


Triangular fractal #6


Triangular fractal #7


Triangular fractal #8


Triangular fractal #9


Triangular fractal #10


Triangular fractal (animated)


Because this fractal is created from a series of star, you could call it a fractar. Here’s a black-and-white version:

Triangular fractar (black-and-white)


Triangular fractar (black-and-white) (animated)
(Open in a new window for larger version if the image seems distorted)


A four-armed star doesn’t yield an easily recognizable fractal in a similar way, so let’s try a five-armed star:

Pentagonal fractar stage #1


Pentagonal fractar #2


Pentagonal fractar #3


Pentagonal fractar #4


Pentagonal fractar #5


Pentagonal fractar #6


Pentagonal fractar #7


Pentagonal fractar (animated)


Pentagonal fractar (black-and-white)


Pentagonal fractar (bw) (animated)


And here’s a six-armed star:

Hexagonal fractar stage #1


Hexagonal fractar #2


Hexagonal fractar #3


Hexagonal fractar #4


Hexagonal fractar #5


Hexagonal fractar #6


Hexagonal fractar (animated)


Hexagonal fractar (black-and-white)


Hexagonal fractar (bw) (animated)


And here’s what happens to the triangular fractar when the new lines are rotated by 60°:

Triangular fractar (60° rotation) #1


Triangular fractar (60°) #2


Triangular fractar (60°) #3


Triangular fractar (60°) #4


Triangular fractar (60°) #5


Triangular fractar (60°) #6


Triangular fractar (60°) #7


Triangular fractar (60°) #8


Triangular fractar (60°) #9


Triangular fractar (60°) (animated)


Triangular fractar (60°) (black-and-white)


Triangular fractar (60°) (bw) (animated)


Triangular fractar (60°) (no lines) (black-and-white)


A four-armed star yields a recognizable fractal when the rotation is 45°:

Square fractar (45°) #1


Square fractar (45°) #2


Square fractar (45°) #3


Square fractar (45°) #4


Square fractar (45°) #5


Square fractar (45°) #6


Square fractar (45°) #7


Square fractar (45°) #8


Square fractar (45°) (animated)


Square fractar (45°) (black-and-white)


Square fractar (45°) (bw) (animated)


Without the lines, the final fractar looks like the plan of a castle:

Square fractar (45°) (bw) (no lines)


And here’s a five-armed star with new lines rotated at 36°:

Pentagonal fractar (36°) #1


Pentagonal fractar (36°) #2


Pentagonal fractar (36°) #3


Pentagonal fractar (36°) #4


Pentagonal fractar (36°) #5


Pentagonal fractar (36°) #6


Pentagonal fractar (36°) #7


Pentagonal fractar (36°) (animated)


Again, the final fractar without lines looks like the plan of a castle:

Pentagonal fractar (36°) (no lines) (black-and-white)


Finally, here’s a six-armed star with new lines rotated at 30°:

Hexagonal fractar (30°) #1


Hexagonal fractar (30°) #2


Hexagonal fractar (30°) #3


Hexagonal fractar (30°) #4


Hexagonal fractar (30°) #5


Hexagonal fractar (30°) #6


Hexagonal fractar (30°) (animated)


And the hexagonal castle plan:

Hexagonal fractar (30°) (black-and-white) (no lines)


Performativizing the Polygonic #2

Suppose a café offers you free drinks for three days. You can have tea or coffee in any order and any number of times. If you want tea every day of the three, you can have it. So here’s a question: how many ways can you choose from two kinds of drink in three days? One simple way is to number each drink, tea = 1, coffee = 2, then count off the choices like this:


1: 111
2: 112
3: 121
4: 122
5: 211
6: 212
7: 221
8: 222

Choice #1 is 111, which means tea every day. Choice #6 is 212, which means coffee on day 1, tea on day 2 and coffee on day 3. Now look at the counting again and the way the numbers change: 111, 112, 121, 122, 211… It’s really base 2 using 1 and 2 rather than 0 and 1. That’s why there are 8 ways to choose two drinks over three days: 8 = 2^3. Next, note that you use the same number of 1s to count the choices as the number of 2s. There are twelve 1s and twelve 2s, because each number has a mirror: 111 has 222, 112 has 221, 121 has 212, and so on.

Now try the number of ways to choose from three kinds of drink (tea, coffee, orange juice) over two days:


11, 12, 13, 21, 22, 23, 31, 32, 33 (c=9)

There are 9 ways to choose, because 9 = 3^2. And each digit, 1, 2, 3, is used exactly six times when you write the choices. Now try the number of ways to choose from three kinds of drink over three days:


111, 112, 113, 121, 122, 123, 131, 132, 133, 211, 212, 213, 221, 222, 223, 231, 232, 233, 311, 312, 313, 321, 322, 323, 331, 332, 333 (c=27)

There are 27 ways and (by coincidence) each digit is used 27 times to write the choices. Now try three drinks over four days:


1111, 1112, 1113, 1121, 1122, 1123, 1131, 1132, 1133, 1211, 1212, 1213, 1221, 1222, 1223, 1231, 1232, 1233, 1311, 1312, 1313, 1321, 1322, 1323, 1331, 1332, 1333, 2111, 2112, 2113, 2121, 2122, 2123, 2131, 2132, 2133, 2211, 2212, 2213, 2221, 2222, 2223, 2231, 2232, 2233, 2311, 2312, 2313, 2321, 2322, 2323, 2331, 2332, 2333, 3111, 3112, 3113, 3121, 3122, 3123, 3131, 3132, 3133, 3211, 3212, 3213, 3221, 3222, 3223, 3231, 3232, 3233, 3311, 3312, 3313, 3321, 3322, 3323, 3331, 3332, 3333 (c=81)

There are 81 ways to choose and each digit is used 108 times. But the numbers don’t have represent choices of drink in a café. How many ways can a point inside an equilateral triangle jump four times half-way towards the vertices of the triangle? It’s the same as the way to choose from three drinks over four days. And because the point jumps toward each vertex in a symmetrical way the same number of times, you get a nice even pattern, like this:

vertices = 3, jump = 1/2


Every time the point jumps half-way towards a particular vertex, its position is marked in a unique colour. The fractal, also known as a Sierpiński triangle, actually represents all possible choices for an indefinite number of jumps. Here’s the same rule applied to a square. There are four vertices, so the point is tracing all possible ways to choose four vertices for an indefinite number of jumps:

v = 4, jump = 1/2


As you can see, it’s not an obvious fractal. But what if the point jumps two-thirds of the way to its target vertex and an extra target is added at the centre of the square? This attractive fractal appears:

v = 4 + central target, jump = 2/3


If the central target is removed and an extra target is added on each side, this fractal appears:

v = 4 + 4 midpoints, jump = 2/3


That fractal is known as a Sierpiński carpet. Now up to the pentagon. This fractal of endlessly nested contingent pentagons is created by a point jumping 1/φ = 0·6180339887… of the distance towards the five vertices:

v = 5, jump = 1/φ


With a central target in the pentagon, this fractal appears:

v = 5 + central, jump = 1/φ


The central red pattern fits exactly inside the five that surround it:

v = 5 + central, jump = 1/φ (closeup)


v = 5 + c, jump = 1/φ (animated)


For a fractal of endlessly nested contingent hexagons, the jump is 2/3:

v = 6, jump = 2/3


With a central target, you get a filled variation of the hexagonal fractal:

v = 6 + c, jump = 2/3


And for a fractal of endlessly nested contingent octagons, the jump is 1/√2 = 0·7071067811… = √½:

v = 8, jump = 1/√2


Previously pre-posted:

Performativizing the Polygonic

Rigging in the Trigging

Here’s a simple pattern of three triangles:

Three-Triangle Pattern


Now replace each triangle in the pattern with the same pattern at a smaller scale:

Replacing triangles


If you keep on doing this, you create what I’ll call a trigonal fractal (trigon is Greek for “triangle”):

Trigonal Fractal stage #3 (click for larger)


Trigonal Fractal stage #4


Trigonal Fractal stage #5


Trigonal Fractal #6


Trigonal Fractal #7


Trigonal Fractal #8


Trigonal Fractal (animated) (click for larger)


You can use the same pattern to create different fractals by rotating the replacement patterns in different ways. I call this “rigging the trigging” and here are some of the results:




You can also use a different seed-pattern to create the fractals:

Trigonal fractal (animated)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)



Trigonal fractal (anim)


Note: The title of this incendiary intervention is of course a paronomasia on the song “Frigging in the Rigging”, also known as “Good Ship Venus” and performed by the Sex Pistols on The Great Rock ’n’ Roll Swindle (1979).