Jumping Jehosophracts!

As I’ve shown pre-previously on Overlord-in-terms-of-issues-around-the-Über-Feral, you can create interesting fractals by placing restrictions on a point jumping inside a fractal towards a randomly chosen vertex. For example, the point can be banned from jumping towards the same vertex twice in a row, and so on.

But you can use other restrictions. For example, suppose that the point can jump only once or twice towards any vertex, that is, (j = 1,2). It can then jump towards the same vertex again, but not the same number of times as it previously jumped. So if it jumps once, it has to jump twice next time; and vice versa. If you use this rule on a pentagon, this fractal appears:

v = 5, j = 1,2 (black-and-white)


v = 5, j = 1,2 (colour)


If the point can also jump towards the centre of the pentagon, this fractal appears:

v = 5, j = 1,2 (with centre)


And if the point can also jump towards the midpoints of the sides:

v = 5, j = 1,2 (with midpoints)


v = 5, j = 1,2 (with midpoints and centre)


And here the point can jump 1, 2 or 3 times, but not once in a row, twice in a row or thrice in a row:

v = 5, j = 1,2,3


v = 5, j = 1,2,3 (with centre)


Here the point remembers its previous two moves, rather than just its previous move:

v = 5, j = 1,2,3, hist = 2 (black-and-white)


v = 5, j = 1,2,3, hist = 2


v = 5, j = 1,2,3, hist = 2 (with center)


v = 5, j = 1,2,3, hist = 2 (with midpoints)


v = 5, j = 1,2,3, hist = 2 (with midpoints and centre)


And here are hexagons using the same rules:

v = 6, j = 1,2 (black-and-white)


v = 6, j = 1,2


v = 6, j = 1,2 (with centre)


And octagons:

v = 8, j = 1,2


v = 8, j = 1,2 (with centre)


v = 8, j = 1,2,3, hist = 2


v = 8, j = 1,2,3, hist = 2


v = 8, j = 1,2,3,4 hist = 3


v = 8, j = 1,2,3,4 hist = 3 (with center)


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The Hex Fractor

Pre-previously on Overlord-in-terms-of-issues-around-the-Über-Feral, I looked at the fractals created when various restrictions are placed on a point jumping at random half-way towards the vertices of a square. For example, the point can be banned from jumping towards the same vertex twice in a row or towards the vertex to the left of the vertex it has just jumped towards, and so on.

Today I want to look at what happens to a similar point moving inside pentagons and hexagons. If the point can’t jump twice towards the same vertex of a pentagon, this is the fractal that appears:

Ban second jump towards same vertex (v + 0)


Ban second jump towards same vertex (color)


If the point can’t jump towards the vertex immediately to the left of the one it’s just jumped towards, this is the fractal that appears:

Ban jump towards v + 1


Ban jump towards v + 1 (color)


And this is the fractal when the ban is on the vertex two places to the left:

Ban jump towards v + 2


Ban jump towards v + 2 (color)


You can also ban more than one vertex:

Ban jump towards v + 0,1


Ban jump towards v + 1,2


Ban jump towards v + 1,4


Ban jump towards v + 1,4 (color)


Ban jump towards v + 2,3


And here are fractals created in similar ways inside hexagons:

Ban jump towards v + 0,1


Ban jump towards v + 0,3


Ban jump towards v + 0,1,2


Ban jump towards v + 0,1,2 (color)


Ban jump towards v + 0,1,4


Ban jump towards v + 0,1,5


Ban jump towards v + 0,2,4


Ban jump towards v + 0,2,4 (color)


Ban jump towards v + 1,2,3


Ban jump towards v + 1,2,3 (color)


Ban jump towards v + 1,2,4


Ban jump towards v + 1,2,4, (color)


Ban jump towards v + 1,3,5


Ban jump towards v + 1,3,5 (color)


Ban jump towards v + 1,2


Ban jump towards v + 1,2


Ban jump towards v + 1,3


Ban jump towards v + 1,3 (color)


Ban jump towards v + 1,5


Ban jump towards v + 1,5 (color)


Ban jump towards v + 2,3


Ban jump towards v + 2,3 (color)


Ban jump towards v + 2,4


Ban jump towards v + 2,4 (color)


Elsewhere other-accessible:

Square Routes Re-Verticed

Square Routes Re-Verticed

Start with a point in the middle of a square. Allow it to make a series of, say, eight jumps towards the vertices of the square, but with one restriction: it can’t jump towards the same vertex twice in a row. When the point has made the eight jumps, mark its position. If you do this for every possible route, the result will look like this:

Ban jump towards same vertex


And here’s a different restriction: the point can’t jump towards the vertex immediately to the left of the vertex it has just jumped towards:

Ban jump towards v + 1


And here it can’t jump towards the vertex diagonally opposite the vertex it has just jumped towards:

Ban jump towards v + 2


Now allow the point to jump not just towards the vertices, but towards points midway between the vertices. And expand and reverse the restrictions: instead of not allowing a jump towards v + i1, v + i2…, only allow a jump towards v + i1, v + i2… Some interesting shapes appear:

Jump must be towards v, v + 1 or v + 2 (one point between vertices)


v, v + 1 or v + 6


v, v + 2 or v + 3


v, v + 2 or v + 4


v, v + 2 or v + 6


v, v + 3 or v + 4


v, v + 3 or v + 5


v, v + 2 or v + 7


v + 1, v + 4 or v + 7


v, v + 1 or v + 6 (two points between vertices)


v, v + 2 or v + 4


v, v + 2 or v + 6


v, v + 2 or v + 9


v, v + 3 or v + 6


v, v + 3 or v + 8


v, v + 4 or v + 8


v, v + 5 or v + 7


v , v + 6 or v + 11


v + 1, v + 5 or v + 6


v + 1, v + 2 or v + 10


v + 1, v + 6 or v + 10


v + 1, v + 6 or v + 11


v + 2, v + 6 or v + 10


Elsewhere other-posted:

Square Routes
Square Routes Revisited
Square Routes Re-Revisited
Square Routes Re-Re-Revisited

Mod’s Chosen

When you divide one integer by another, one of two things happens. Either the second number goes perfectly into the first or there’s a remainder:


15 / 5 = 3
18 / 5 = 3⅗

In the first case, there’s no remainder, that is, the remainder is 0. In the second case, there’s a remainder of 3. And all that gives you the basis for what’s called modular arithmetic. It returns the remainder when one number is divided by another:


15 mod 5 = 0
16 mod 5 = 1
17 mod 5 = 2
18 mod 5 = 3
19 mod 5 = 4
20 mod 5 = 0
21 mod 5 = 1
22 mod 5 = 2...

It looks simple but a lot of mathematics is built on it. I don’t know much of that maths, but I know one thing I like: the patterns you can get from modular arithmetic. Suppose you draw a square, then find a point and measure the distances from that point to all the vertices of the square. Then add the distances up, turn the result into an integer if necessary, and test whether the result is divisible by 2 or not. If it is divisible, colour the point in. If it isn’t, leave the point blank.

Then move on to another point and perform the same test. This is modular arithematic, because for each point you’re asking whether d mod 2 = 0. The result looks like this:

d mod 2 = 0


Here are more divisors:

d mod 3 = 0


d mod 4 = 0


d mod 5 = 0


d mod 6 = 0


d mod 7 = 0


d mod 8 = 0


d mod 9 = 0


d mod 10 = 0


d mod various = 0 (animated)


You can also use modular arithmetic to determine the colour of the points. For example, if d mod n = 0, the point is black; if d mod n = 1, the point is red; if d mod n = 2, the point is green; and so on.

d mod 3 = 0, 1, 2 (coloured)


d mod 4 = 0, 1, 2, 3 (coloured)


d mod 5 = 0, 1, 2, 3, 4 (coloured)



d mod 5 = 0, 1, 2, 3, 4 (animated and expanding)


Bat out of L

Pre-previously on Overlord-in-terms-of-the-Über-Feral, I’ve looked at intensively interrogated issues around the L-triomino, a shape created from three squares that can be divided into four copies of itself:

An L-triomino divided into four copies of itself


I’ve also interrogated issues around a shape that yields a bat-like fractal:

A fractal full of bats


Bat-fractal (animated)


Now, to end the year in spectacular fashion, I want to combine the two concepts pre-previously interrogated on Overlord-in-terms-of-the-Über-Feral (i.e., L-triominoes and bats). The L-triomino can also be divided into nine copies of itself:

An L-triomino divided into nine copies of itself


If three of these copies are discarded and each of the remaining six sub-copies is sub-sub-divided again and again, this is what happens:

Fractal stage 1


Fractal stage 2


Fractal #3


Fractal #4


Fractal #5


Fractal #6


Et voilà, another bat-like fractal:

L-triomino bat-fractal (static)


L-triomino bat-fractal (animated)


Elsewhere other-posted:

Tri-Way to L
Bats and Butterflies
Square Routes
Square Routes Revisited
Square Routes Re-Revisited
Square Routes Re-Re-Revisited

Tridentine Math

The Tridentine Mass is the Roman Rite Mass that appears in typical editions of the Roman Missal published from 1570 to 1962. — Tridentine Mass, Wikipedia

A 30°-60°-90° right triangle, with sides 1 : √3 : 2, can be divided into three identical copies of itself:

30°-60°-90° Right Triangle — a rep-3 rep-tile…


And if it can be divided into three, it can be divided into nine:

…that is also a rep-9 rep-tile


Five of the sub-copies serve as the seed for an interesting fractal:

Fractal stage #1


Fractal stage #2


Fractal stage #3


Fractal #4


Fractal #5


Fractal #6


Fractal #6


Tridentine cross (animated)


Tridentine cross (static)


This is a different kind of rep-tile:

Noniamond trapezoid


But it yields the same fractal cross:

Fractal #1


Fractal #2


Fractal #3


Fractal #4


Fractal #5


Fractal #6


Tridentine cross (animated)


Tridentine cross (static)


Elsewhere other-available:

Holey Trimmetry — another fractal cross

Square Routes Re-Re-Revisited

This is an L-triomino, or shape created from three squares laid edge-to-edge:

When you divide each square like this…

You can create a fractal like this…

Stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Stage #9


Stage #10


Animated fractal


Here are more fractals created from the triomino:

Animated


Static


Animated


Static


Animated


Static


And here is a different shape created from three squares:

And some fractals created from it:

Animated


Static


Animated


Static


Animated


Static


And a third shape created from three squares:

And some fractals created from it:

Animated


Static


Animated


Static


Animated


Static


Animated


Static


Animated


Static


Animated


Static


Animated


Static


Animated


Static


Previously pre-posted (please peruse):

Tri-Way to L
Square Routes
Square Routes Revisited
Square Routes Re-Revisited

Bats and Butterflies

I’ve used butterfly-images to create fractals. Now I’ve found a butterfly-image in a fractal. The exciting story begins with a triabolo, or shape created from three isoceles right triangles:


The triabolo is a rep-tile, or shape that can be divided into smaller copies of itself:


In this case, it’s a rep-9 rep-tile, divisible into nine smaller copies of itself. And each copy can be divided in turn:


But what happens when you sub-divide, then discard copies? A fractal happens:

Fractal crosses (animated)


Fractal crosses (static)


That’s a simple example; here is a more complex one:

Fractal butterflies #1


Fractal butterflies #2


Fractal butterflies #3


Fractal butterflies #4


Fractal butterflies #5


Fractal butterflies (animated)


Some of the gaps in the fractal look like butterflies (or maybe large moths). And each butterfly is escorted by four smaller butterflies. Another fractal has gaps that look like bats escorted by smaller bats:

Fractal bats (animated)

Fractal bats (static)


Elsewhere other-posted:

Gif Me Lepidoptera — fractals using butterflies
Holey Trimmetry — more fractal crosses

Holey Trimmetry

Symmetry arising from symmetry isn’t surprising. But what about symmetry arising from asymmetry? You can find both among the rep-tiles, which are geometrical shapes that can be completely replaced by smaller copies of themselves. A square is a symmetrical rep-tile. It can be replaced by nine smaller copies of itself:

Rep-9 Square

If you trim the copies so that only five are left, you have a symmetrical seed for a symmetrical fractal:

Fractal cross stage #1


Fractal cross #2


Fractal cross #3


Fractal cross #4


Fractal cross #5


Fractal cross #6


Fractal cross (animated)


Fractal cross (static)


If you trim the copies so that six are left, you have another symmetrical seed for a symmetrical fractal:

Fractal Hex-Ring #1


Fractal Hex-Ring #2


Fractal Hex-Ring #3


Fractal Hex-Ring #4


Fractal Hex-Ring #5


Fractal Hex-Ring #6


Fractal Hex-Ring (animated)


Fractal Hex-Ring (static)


Now here’s an asymmetrical rep-tile, a nonomino or shape created from nine squares joined edge-to-edge:

Nonomino


It can be divided into twelve smaller copies of itself, like this:

Rep-12 Nonomino (discovered by Erich Friedman)


If you trim the copies so that only five are left, you have an asymmetrical seed for a familiar symmetrical fractal:

Fractal cross stage #1


Fractal cross #2


Fractal cross #3


Fractal cross #4


Fractal cross #5


Fractal cross #6


Fractal cross (animated)


Fractal cross (static)


If you trim the copies so that six are left, you have an asymmetrical seed for another familiar symmetrical fractal:

Fractal Hex-Ring #1


Fractal Hex-Ring #2


Fractal Hex-Ring #3


Fractal Hex-Ring #4


Fractal Hex-Ring #5


Fractal Hex-Ring (animated)


Fractal Hex-Ring (static)


Elsewhere other-available:

Square Routes Re-Re-Visited

Bent for the Pent

A triangle can be tiled with triangles and a square with squares, but a pentagon can’t be tiled with pentagons. At least, not in the same way, using smaller copies of the same shape. The closest you can get is this:

Pentaflake #1


If you further subdivide the pentagon, you create what is known as a pentaflake:

Pentaflake #2


Pentaflake #3


Pentaflake #4


Pentaflake (animated)


Pentaflake (static)


But if you bend the rules and use irregular smaller pentagons, you can tile a pentagon like this, creating what I called a pentatile:

Pentatile stage 1


Further subdivisions create an interesting final pattern:

Pentatile #2


Pentatile #3


Pentatile #4


Pentatile #5


Pentatile #6


Pentatile (animated)


Pentatile (static)


By varying the size of the central pentagon, you can create other patterns:

Pentatile #1 (animated)


Pentatile #2 (animated)

Pentatile #2







Pentatile with no central pentagon


And here are various pentatiles in an animated gif:


And here are some variations on the pentaflake:







Elsewhere other-posted:

Bent for the Rent (1976) — the title of the incendiary intervention above is of course a reference to the “first and last glitter-rock album” by England’s loudest band, Spinal In Terms Of Tap
Phrallic Frolics — more on pentaflakes