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)


Koch Rock

The Koch snowflake, named after the Swedish mathematician Helge von Koch, is a famous fractal that encloses a finite area within an infinitely long boundary. To make a ’flake, you start with an equilateral triangle:

Koch snowflake stage #1 (with room for manœuvre)


Next, you divide each side in three and erect a smaller equilateral triangle on the middle third, like this:

Koch snowflake #2


Each original straight side of the triangle is now 1/3 longer, so the full perimeter has also increased by 1/3. In other words, perimeter = perimeter * 1⅓. If the perimeter of the equilateral triangle was 3, the perimeter of the nascent Koch snowflake is 4 = 3 * 1⅓. The area of the original triangle also increases by 1/3, because each new equalitarian triangle is 1/9 the size of the original and there are three of them: 1/9 * 3 = 1/3.

Now here’s stage 3 of the snowflake:

Koch snowflake #3, perimeter = 4 * 1⅓ = 5⅓


Again, each straight line on the perimeter has been divided in three and capped with a smaller equilateral triangle. This increases the length of each line by 1/3 and so increases the full perimeter by a third. 4 * 1⅓ = 5⅓. However, the area does not increase by 1/3. There are twelve straight lines in the new perimeter, so twelve new equilateral triangles are erected. However, because their sides are 1/9 as long as the original side of the triangle, they have 1/(9^2) = 1/81 the area of the original triangle. 1/81 * 12 = 4/27 = 0.148…

Koch snowflake #4, perimeter = 7.11


Koch snowflake #5, p = 9.48


Koch snowflake #6, p = 12.64


Koch snowflake #7, p = 16.85


Koch snowflake (animated)


The perimeter of the triangle increases by 1⅓ each time, while the area reaches a fixed limit. And that’s how the Koch snowflake contains a finite area within an infinite boundary. But the Koch snowflake isn’t confined to itself, as it were. In “Dissecting the Diamond”, I described how dissecting and discarding parts of a certain kind of diamond could generate one side of a Koch snowflake. But now I realize that Koch snowflakes are everywhere in the diamond — it’s a Koch rock. To see how, let’s start with the full diamond. It can be divided, or dissected, into five smaller versions of itself:

Dissectable diamond


When the diamond is dissected and three of the sub-diamonds are discarded, two sub-diamonds remain. Let’s call them sub-diamonds 1 and 2. When this dissection-and-discarding is repeated again and again, a familiar shape begins to appear:

Koch rock stage 1


Koch rock #2


Koch rock #3


Koch rock #4


Koch rock #5


Koch rock #6


Koch rock #7


Koch rock #8


Koch rock #9


Koch rock #10


Koch rock #11


Koch rock #12


Koch rock #13


Koch rock (animated)


Dissecting and discarding the diamond creates one side of a Koch triangle. Now see what happens when discarding is delayed and sub-diamonds 1 and 2 are allowed to appear in other parts of the diamond. Here again is the dissectable diamond:

Dia-flake stage 1


If no sub-diamonds are discarded after dissection, the full diamond looks like this when each sub-diamond is dissected in its turn:

Dia-flake #2


Now let’s start discarding sub-diamonds:

Dia-flake #3


And now discard everything but sub-diamonds 1 and 2:

Dia-flake #4


Dia-flake #5


Dia-flake #6


Dia-flake #7


Dia-flake #8


Dia-flake #9


Dia-flake #10


Now full Koch snowflakes have appeared inside the diamond — count ’em! I see seven full ’flakes:

Dia-flake #11


Dia-flake (animated)


But that isn’t the limit. In fact, an infinite number of full ’flakes appear inside the diamond — it truly is a Koch rock. Here are examples of how to find more full ’flakes:

Dia-flake 2 (static)


Dia-flake 2 (animated)


Dia-flake 3 (static)


Dia-flake 3 (animated)


Previously pre-posted:

Dissecting the Diamond — other fractals in the dissectable diamond

Dissecting the Diamond

Pre-previously on O.o.t.Ü.-F., I dilated the delta. Now I want to dissect the diamond. In geometry, a shape is dissected when it is completely divided into smaller shapes of some kind. If the smaller shapes are identical (except for size) to the original, the original shape is called a rep-tile (because it can be tiled with repeating versions of itself). If the smaller identical shapes are equal in size to each other, the rep-tile is regular; if the smaller shapes are not equal, the rep-tile is irregular. This diamond is an irregular rep-tile or irrep-tile:

Dissectable diamond

Dissected diamond


As you can see, the diamond can be dissected into five smaller versions of itself, two larger ones and three smaller ones. This makes it a rep-5 irrep-tile. And the smaller versions, or sub-diamonds, can themselves be dissected ad infinitum, like this:

Dissected diamond stage #1


Dissected diamond #2


Dissected diamond #3


Dissected diamond #4


Dissected diamond #5


Dissected diamond #6


Dissected diamond #7


Dissected diamond #8


Dissected diamond #9


Dissected diamond (animated)


The full dissected diamond is a fractal, or shape that is similar to itself at varying scales. However, the fractality of the diamond becomes most obvious when you dissect-and-discard. That is, first you dissect the diamond, then you discard one (or more) of the sub-diamonds, like this:

Diamond fractal (retaining sub-diamonds 1,2,3,4) stage #1


1234-Diamond #2


1234-Diamond #3


1234-Diamond #4


1234-Diamond #5


1234-Diamond #6


1234-Diamond #7


1234-Diamond #8


1234-Diamond #9


1234-Diamond (animated)


Here are some more fractals created by dissecting and discarding one sub-diamond:

Diamond fractal (retaining sub-diamonds 1,2,4,5)


1245-Diamond (anim)


2345-Diamond


2345-Diamond (anim)


The 2345-diamond fractal has variants created by mirroring one or more sub-diamonds, so that the orientation of the sub-dissections changes. Here is one of the variants:

2345-Diamond (variation)


2345-Diamond (variant) (anim)


And here is a fractal created by dissecting and discarding two sub-diamonds:

Diamond fractal (retaining sub-diamonds 1,2,3)


123-Diamond (anim)


Again, the fractal has variants created by mirroring one or more of the sub-diamonds:

123-Diamond (variant #1)


123-Diamond (variant #2)


123-Diamond (variant #3)


123-Diamond (variant #4)


Some more fractals created by dissecting and discarding two sub-diamonds:

125-Diamond


125-Diamond (anim)


134-Diamond


134-Diamond (anim)


235-Diamond


235-Diamond (anim)


135-Diamond


135-Diamond (anim)


A variant of the 135-Diamond fractal looks like one side of a Koch snowflake:

135-Diamond (variant #1) — like Koch snowflake


135-Diamond (variant #2)


Finally, here are some colour variants of the full dissected diamond:






Full diamond colour variants (anim)


Elsewhere other-engageable:

Dilating the Delta

Curvous Energy

Here is a strange and beautiful fractal known as a dragon curve:

A dragon curve (note: this is a twin-dragon curve or Davis-Knuth dragon)


And here is the shape generally regarded as the dullest and most everyday of all:

A square


But squares are square, so let’s go back to dragon-curves. This particular kind of dragon-curve looks a lot like a Chinese dragon. You can see the same writhing energy and scaliness:

Chinese dragon


Dragon-curve for comparison


Dragon-curves also look like some species of soft coral:

Red soft-coral


In short, dragon-curves are organic and lively, quite unlike the rigid, lifeless solidity of a square. But there’s more to a dragon-curve than immediately meets the eye. Dragon-curves are rep-tiles, that is, you can tile one with smaller copies of itself:

Dragon-curve rep-tiled with two copies of itself


Dragon-curve rep-4


Dragon-curve rep-8


Dragon-curve rep-16


Dragon-curve rep-32


Dragon-curve self-tiling (animated)


From the rep-32 dragon-curve, you can see that a dragon-curve can be surrounded by six copies of itself. Here’s an animation of the process:

Dragon-curve surrounded (anim)


And because dragon-curves are rep-tiles, they will tile the plane:

Dragon-curve tiling #1


Dragon-curve tiling #2


But how do you make these strange and beautiful shapes, with their myriad curves and curlicules, their energy and liveliness? It’s actually very simple. You start with the shape generally regarded as the dullest and most everyday of all:

A square


Then you see how the shape can be replaced by five smaller copies of itself:

Square overlaid by five smaller squares


Square replaced by five smaller squares


Then you set about replacing it with two of those smaller copies:

Replacing squares Stage #0


Replacing squares Stage #1


Then you do it again to each of the copies:

Replacing squares Stage #2


And again:

Replacing squares #3


And again:

Replacing squares #4


And keep on doing it:

Replacing squares #5


Replacing squares #6


Replacing squares #7


Replacing squares #8


Replacing squares #9


Replacing squares #10


Replacing squares #11


Replacing squares #12


Replacing squares #13


Replacing squares #14


Replacing squares #15


And in the end you’ve got a dragon-curve:

Dragon-curve built from squares


Dragon-curve built from squares (animated)


Square Routes Re-Re-Re-Revisited

Discovering something that’s new to you in recreational maths is good. But so is re-discovering it by a different route. I’ve long been passionate about what happens when a point is allowed to jump repeatedly halfway towards the randomly chosen vertices of a square. If the point can choose any vertex any number of times, the interior of the square fills slowly and completely with points, like this:

Point jumping at random halfway towards vertices of a square


However, if the point is banned from jumping towards the same vertex twice or more in a row, an interesting fractal appears:

Fractal #1 — ban on jumping towards vertex vi twice or more


If the point can’t jump towards the vertex one place clockwise of the vertex it’s just jumped towards, this fractal appears:

Fractal #2 — ban on jumping towards vertex vi+1


If the point can’t jump towards the vertex two places clockwise of the vertex it’s just jumped towards, this fractal appears (two places clockwise is also two places anticlockwise, i.e. the banned vertex is diagonally opposite):

Fractal #3 — ban on jumping towards vertex vi+2


Now I’ve discovered a new way to create these fractals. You take a filled square, divide it into smaller squares, then remove some of them in a systematic way. Then you do the same to the smaller squares that remain. For fractal #1, you do this:

Fractal #1, stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Fractal #1 (animated)


For fractal #2, you do this:

Fractal #2, stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Fractal #2 (animated)


For fractal #3, you do this:

Fractal #3, stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Fractal #3 (animated)


If the sub-squares are coloured, it’s easier to understand how, say, fractal #1 is created:

Fractal #1 (coloured), stage #1


Stage #2


Stage #3


Stage #4


Stage #5


Stage #6


Stage #7


Stage #8


Fractal #1 (coloured and animated)


The fractal is actually being created in quarters, with one quarter rotated to form the second, third and fourth quarters:

Fractal #1, quarter









Here’s an animation of the same process for fractal #3:

Fractal #3 (coloured and animated)


So you can create these fractals either with a jumping point or by subdividing a square. But in fact I discovered the subdivided-square route by looking at a variant of the jumping-point route. I wondered what would happen if you took a point inside a square, allowed it to trace all possible routes towards the vertices without marking its position, then imposed the restriction for Fractal #1 on its final jump, namely, that it couldn’t jump towards the vertex it jumped towards on its previous jump. If the point is marked after its final jump, this is what appears (if the routes chosen had been truly random, the image would be similar but messier):

Fractal #1, restriction on final jump


Then I imposed the same restriction on the point’s final two jumps:

Fractal #1, restriction on final 2 jumps


And final three jumps:

Fractal #1, restriction on final 3 jumps


And so on:

Fractal #1, restriction on final 4 jumps


Fractal #1, restriction on final 5 jumps


Fractal #1, restriction on final 6 jumps


Fractal #1, restriction on final 7 jumps


Here are animations of the same process applied to fractals #2 and #3:

Fractal #2, restrictions on final 1, 2, 3… jumps


Fractal #3, restrictions on final 1, 2, 3… jumps


The longer the points are allowed to jump before the final restriction is imposed on their n final jumps, the more densely packed the marked points will be:

Fractal #1, packed points #1


Packed points #2


Packed points #3


Eventually, the individual points will form a solid mass, like this:

Fractal #1, solid mass of points


Fractal #1, packed points (animated)


Previously pre-posted (please peruse):

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

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

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