Monthly Archives: August 2019

Square Routes Re-Re-Re-Re-Revisited

by in Fractals, Geometry, Mathematics, Squares and tagged , , , , , , , , , , , , | Leave a comment

Pre-previously in my post-passionate portrayal of polygonic performativity, I’ve usually looked at what happens when a moving point is banned from jumping twice-in-a-row (and so on) towards the same vertex of a square or other polygon. But what happens when the point isn’t banned but compelled to do something different? For example, if the point usually jumps 1/2 of the distance towards the vertex for the second (third, fourth…) time, you could make it jump 2/3 of the way, like this:

usual jump = 1/2, forced jump = 2/3

And here are the fractals created when the vertex currently chosen is one or two places clockwise from the vertex chosen before:

usual jump = 1/2, forced jump = 2/3, vertex-inc = +1

j1 = 1/2, j2 = 2/3, vi = +2

Or you can make the point jump towards a different vertex to the one chosen, without recording the different vertex in the history of jumps:

v1 = +0, v2 = +1, j = 1/2

v1 = +0, v2 = +1, vi = +2

v1 = 0, v2 = +2

v1 = 0, v2 = +2, vi = +1

Or you can make the point jump towards the center of the square:

v1 = 0, v2 = center, j = 1/2

v1 = 0, v2 = center, vertex-inc = +1

v1 = 0, v2 = center, vertex-inc = +2

And so on:

v1 = +1, v2 = +1, vi = +1

v1 = +1, v2 = +1, vi = +2

v1 = +0, v2 = +1, reverse test

v1 = +0, v2 = +1, vi = +1, reverse test

v1 = +0, v2 = +1, vi = +2, reverse test

v1 = +0, v2 = +2, reverse test

v1 = +0, v2 = +2, vi = +1, reverse test

v1 = +2, v2 = +2, vi = +1, reverse test

j1 = 1/2, j2 = 2/3, vi = +0,+0 (record previous two jumps in history)

j1 = 1/2, j2 = 2/3, vi = +0,+2

j1 = 1/2, j2 = 2/3, vi = +2,+2

j1 = 1/2, j2 = 2/3, vi = +0,+0,+0 (previous three jumps)

Previously pre-posted (please peruse):

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

Lost Lustre

by in Biology, Entomology, Lepidoptera, Natural History, Quotations and tagged , , , , , , , , , , , , | Leave a comment

“Adonis, M. Cytheris, and M. Menelaus, is indescribable; the eyes are pained as they gaze upon it; yet there is said to be an unnamed species from the emerald mountains of Bogota, of which a single specimen is in a private cabinet in London, which is far more lustrous than these.” — The Romance of Natural History (1861), Philip Henry Gosse

Monbiot’s Mothbiota

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When they opened the trap, I was astonished by the range and beauty of their catch. There were pink and olive elephant hawkmoths; a pine hawkmoth, feathered and ashy; a buff arches, patterned and gilded like the back of a barn owl; flame moths in polished brass; the yellow kites of swallow-tailed moths; common emeralds the colour of a northern sea, with streaks of foam; grey daggers; a pebble prominent; heart and darts; coronets; riband waves; willow beauties; an elder pearl; small magpie; double-striped pug; rosy tabby. The names testify to a rich relationship between these creatures and those who love them. — George Monbiot, “Our selective blindness is lethal to the living world”, The Guardian, 20xii2017

Get Your Prox Off #3

by in Fractals, Geometry, Mathematics and tagged , , , , , , , , , , , , , , | Leave a comment

I’ve looked at lot at the fractals created when you randomly (or quasi-randomly) choose a vertex of a square, then jump half of the distance towards it. You can ban jumps towards the same vertex twice in a row, or jumps towards the vertex clockwise or anticlockwise from the vertex you’ve just chosen, and so on.

But you don’t have to choose vertices directly: you can also choose them by distance or proximity (see “Get Your Prox Off” for an earlier look at fractals-by-distance). For example, this fractal appears when you can jump half-way towards the nearest vertex, the second-nearest vertex, and the third-nearest vertex (i.e., you can’t jump towards the fourth-nearest or most distant vertex):

vertices = 4, distance = (1,2,3), jump = 1/2

It’s actually the same fractal as you get when you choose vertices directly and ban jumps towards the vertex diagonally opposite from the one you’ve just chosen. But this fractal-by-distance isn’t easy to match with a fractal-by-vertex:

v = 4, d = (1,2,4), j = 1/2

Nor is this one:

v = 4, d = (1,3,4)

This one, however, is the same as the fractal-by-vertex created by banning a jump towards the same vertex twice in a row:

v = 4, d = (2,3,4)

The point can jump towards second-nearest, third-nearest and fourth-nearest vertices, but not towards the nearest. And the nearest vertex will be the one chosen previously.

Now let’s try squares with an additional point-for-jumping-towards on each side (the points are numbered 1 to 8, with points 1, 3, 5, 7 being the true vertices):

v = 4 + s1 point on each side, d = (1,2,3)

v = 4 + s1, d = (1,2,5)

v = 4 + s1, d = (1,2,7)

v = 4 + s1, d = (1,3,8)

v = 4 + s1, d = (1,4,6)

v = 4 + s1, d = (1,7,8)

v = 4 + s1, d = (2,3,8)

v = 4 + s1, d = (2,4,8)

And here are squares where the jump is 2/3, not 1/2, and you can choose only the nearest or third-nearest jump-point:

v = 4, d = (1,3), j = 2/3

v = 4 + s1, d = (1,3), j = 2/3

Now here are some pentagonal fractals-by-distance:

v = 5, d = (1,2,5), j = 1/2

v = 5 + s1, d = (1,2,7)

v = 5 + s1, d = (1,2,8)

v = 5 + s1, d = (1,2,9)

v = 5 + s1, d = (1,9,10)

v = 5 + s1, d = (1,10), j = 2/3

v = 5 + s1, d = (various), j = 2/3 (animated)

And now some hexagonal fractals-by-distance:

v = 6, d = (1,2,4), j = 1/2

v = 6, d = (1,3,5)

v = 6, d = (1,3,6)

v = 6, d = (1,2,3,4)

v = 6 + central point, d = (1,2,3,4)

v = 6, d = (1,2,3,6)

v = 6, d = (1,2,4,6)

v = 6, d = (1,3,4,5)

v = 6, d = (1,3,4,6)

v = 6, d = (1,4,5,6)

Elsewhere other-accessible:

• Get Your Prox Off — an earlier look at fractals-by-distance
• Get Your Prox Off # 2 — and another