Blancmange Butterfly

Friday, 29th Sep, 2023 by Krilling for Company in Blancmange Curves, Circles, Fractals, Geometry, Mathematics and tagged animated gifs, animated gifs of fractals, arching blancmange circle, arching blancmange curve, arcs, blancmange circles, blancmange fractal, Circles, curves, fractal butterfly, fractal geometry, fractals, geometry, recreational math, recreational mathematics, recreational maths | Leave a comment

Blancmange butterfly. Is that a ’60s psychedelic band? No, it’s one of the shapes you can get by playing with blancmange curves. As I described in “White Rites”, a blancmange curve is a fractal created by summing the heights of successively smaller and more numerous zigzags, like this:

Zigzags 1 to 10

Zigzags 1 to 10 (animated)

Blancmange curve

In the blancmange curves below, the height (i.e., the y co-ordinate) has been normalized so that all the images are the same height:

Construction of a normalized blancmange curve (animated)

This is the solid version:

Solid normalized blancmange curve (animated)

I wondered what happens when you wrap a blancmange curve around a circle. Well, this happens:

Construction of a blancmange circle (animated)

You get what might be called a blancmange butterfly. The solid version looks like this (patterns in the circles are artefacts of the graphics program I used):

Solid blancmange circle (animated)

Next I tried using arcs rather zigzags to construct the blancmange curves and blancmange circles:

Arching blancmange curve (i.e., constructed with arcs) (animated)

And below is the circular version of a blancmange curve constructed with arcs. The arching circular blancmanges look even more like buttocks and then intestinal villi (the fingerlike projections lining our intestines):

Arching blancmange circle (animated)

The variations on blancmange curves don’t stop there — in fact, they’re infinite. Below is a negative arching blancmange curve, where the heights of the original arching blancmange curve are subtracted from the (normalized) maximum height:

Negative arching blancmange curve (animated)

And here’s an arching blancmange curve that’s alternately negative and positive:

Negative-positive arching blancmange curve (animated)

The circular version looks like this:

Negative-positive arching blancmange circle (animated)

Finally, here’s an arching blancmange curve that’s alternately positive and negative:

Positive-negative arching blancmange curve (animated)

And the circular version:

Positive-negative arching blancmange circle (animated)

Elsewhere Other-Accessible…

• White Rites — more variations on blancmange curves

Mice Thrice

Friday, 6th Jul, 2018 by Krilling for Company in Geometry, Mathematics, Pursuit Curves and tagged animated gif. animated gifs, beetles, curves, geometry, mathematics, maths, mice problem, mouse problem, Pursuit Curves, recreational math, recreational mathematics, recreational maths, Squares | Leave a comment

Twice before on Overlord-in-terms-of-Core-Issues-around-Maximal-Engagement-with-Key-Notions-of-the-Über-Feral, I’ve interrogated issues around pursuit curves. Imagine four mice or four beetles each sitting on one corner of a square and looking towards the centre of the square. If each mouse or beetle begins to run towards the mouse or beetle to its left, it will follow a curving path that takes it to the centre of the square, like this:

vertices = 4, pursuit = +1

The paths followed by the mice or beetles are pursuit curves. If you arrange eight mice clockwise around a square, with a mouse on each corner and a mouse midway along each side, you get a different set of pursuit curves:

v = 4 + 1 on the side, p = +1

Here each mouse is pursuing the mouse two places to its left:

v = 4+s1, p = +2

And here each mouse is pursuing the mouse three places to its left:

v = 4+s1, p = +3

Now try a different arrangement of the mice. In the square below, the mice are arranged clockwise in rows from the bottom right-hand corner. That is, mouse #1 begins on the bottom left-hand corner, mouse #2 begins between that corner and the centre, mouse #3 begins on the bottom left-hand corner, and so on. When each mice runs towards the mouse three places away, these pursuit curves appear:

v = 4 + 1 internally, p = +3

Here are some more:

v = 4 + i1, p = +5

v = 4 + i2, p = +1

v = 4 + i2, p = +2

So far, all the mice have eventually run to the centre of the square, but that doesn’t happen here:

v = 4 + i2, p = 4

Here are more pursuit curves for the v4+i2 mice, using an animated gif:

v = 4 + i2, p = various (animated — open in new tab for clearer image)

And here are more pursuit curves that don’t end in the centre of the square:

v = 4 + i4, p = 4

v = 4 + i4, p = 8

v = 4 + i4, p = 12

v = 4 + i4, p = 16

But the v4+i4 pursuit curves more usually look like this:

v = 4 + i4, p = 7

Now try adapting the rules so that mice don’t run directly towards another mouse, but towards the point midway between two other mice. In this square, the odd- and even-numbered mice follow different rules. Mice #1, #3, #5 and #7 run towards the point midway between the mice one and two places away, while ice #2, #4, #6 and #8 run towards the point midway between the mice two and seven places away:

v = 4 + s1, p(1,3,5,7) = 1,2, p(2,4,6,8) = 2,7

I think the curves are very elegant. Here’s a slight variation:

v = 4 + s1, p1 = 1,3, p2 = 2,7

Now try solid curves:

v = 4 + s1, p1 = 1,3, p2 = 2,7 (red)

v = 4 + s1, p1 = 1,3, p2 = 2,7 (yellow-and-blue)

And some variants:

v = 4 + s1, p1 = 1,7, p2 = 1,2

v = 4 + s1, p1 = 2,3, p2 = 2,5

v = 4 + s1, p1 = 5,6, p2 = 1,3

v = 4 + s1, p1 = 5,6, p2 = 1,4

v = 4 + s1, p1 = 5,6, p2 = 1,6

Elsewhere other-posted:

• Polymorphous Pursuit
• Persecution Complex

Lette’s Roll

Thursday, 7th May, 2015 by Krilling for Company in Geometry, π, Mathematics, Pi and tagged animated gif, animated gifs, cardioid, Circles, curves, family of curves, geometry, math, mathematics, maths, nephroid, paths traced by rolling circles, point on a rolling circle, recreational math, recreational mathematics, recreational maths, rolling circles, roulette, roulettes, wheels | Leave a comment

A roulette is a little wheel or little roller, but it’s much more than a game in a casino. It can also be one of a family of curves created by tracing the path of a point on a rotating circle. Suppose a circle rolls around another circle of the same size. This is the resultant roulette:

The shape is called a cardioid, because it looks like a heart (kardia in Greek). Now here’s a circle with radius r rolling around a circle with radius 2r:

That shape is a nephroid, because it looks like a kidney (nephros in Greek).

This is a circle with radius r rolling around a circle with radius 3r:

And this is r and 4r:

The shapes above might be called outer roulettes. But what if a circle rolls inside another circle? Here’s an inner roulette whose radius is three-fifths (0.6) x the radius of its rollee:

The same roulette appears inverted when the inner circle has a radius two-fifths (0.4) x the radius of the rollee:

But what happens when the circle rolling “inside” is larger than the rollee? That is, when the rolling circle is effectively swinging around the rollee, like a bunch of keys being twirled on an index finger? If the rolling radius is 1.5 times larger, the roulette looks like this:

If the rolling radius is 2 times larger, the roulette looks like this:

Here are more outer, inner and over-sized roulettes:

And you can have circles rolling inside circles inside circles:

And here’s another circle-in-a-circle in a circle:

Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral

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Tag Archives: curves

Blancmange Butterfly

Mice Thrice

Lette’s Roll