The Choice of the Circle

Saturday, 26th Sep, 2015 by Krilling for Company in Geometry, Mathematics and tagged animated gif, animated gifs, circle, Circles, combinations, factorial, factorial n, factorials, graphics, math, mathematics, maths, n!, partitions, points around a circle, Polygons, recreational math, recreational mathematics, recreational maths | Leave a comment

Here’s an elementary mathematical problem: how many ways are there to choose three numbers from a set of six numbers? If the set is (1, 2, 3, 4, 5, 6), these are the possible choices (or combinations):

(1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6), (1, 3, 4), (1, 3, 5), (1, 3, 6), (1, 4, 5), (1, 4, 6), (1, 5, 6), (2, 3, 4), (2, 3, 5), (2, 3, 6), (2, 4, 5), (2, 4, 6), (2, 5, 6), (3, 4, 5), (3, 4, 6), (3, 5, 6), (4, 5, 6) (c = 20)

So 6C3 = 20 (C stands for “combination”). The general formula is nCr = (n! / (n-r)!) / r!, where n is the number to choose from, r is the number of choices and n! is factorial n, or n multiplied by all numbers less than itself. For example, 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720. When n = 6 and c = 3, 6C3 = (6! / (6-3)!) / 3! = (720 / 6) / 6 = 20.

There isn’t much visual appeal in the choices above, but there’s a simple way to change that. Take the ways of choosing two numbers from a set of ten. They start like this:

(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (1, 9), (1, 10), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (2, 8), (2, 9), (2, 10), (3, 4), (3, 5), (3, 6)…

Suppose each choice represents the midpoint of two points chosen from a set of ten points around a pentagon, so that (1, 2) is half-way between points 1 and 2, (3, 5) is half-way between points 3 and 5, and so on:

Now take the ways of choosing three numbers from a set of ten:

(1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6), (1, 2, 7), (1, 2, 8), (1, 2, 9), (1, 2, 10), (1, 3, 4), (1, 3, 5), (1, 3, 6), (1, 3, 7), (1, 3, 8), (1, 3, 9), (1, 3, 10)…

Now the pentagon looks like this, with (1, 2, 3) representing the point midway between 1, 2 and 3, (1, 3, 9) representing the point midway between 1, 3 and 9, and so on:

Now here are 10C4, 10C5 and 10C6 for the pentagon:

You can also generate the points 5C4 = 5, then add them to the original five points and generate 10C4:

5C4

10C4

And here are 5C5, 6C5 and 12C5:

Here are 7C7 and 8C8, adding points as for 5C4:

And here is 12C6 using a dodecagon:

And various nCr for dodecagons and other polygons:

This method can also be used to represent the partitions of n, or the number of sets whose members sum to n. The partitions of 5 are these:

(5), (4, 1), (3, 2), (3, 1, 1), (2, 2, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1)

There are seven partitions, so p(5) = 7. Partitions start small and get very large, starting with p(1), p(2), p(3) and so on:

1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525, 204226, 239943, 281589, 329931, 386155, 451276, 526823, 614154, 715220, 831820, 966467, 1121505, 1300156…

Suppose the partitions of n are treated as sets of points around a polygon with n vertices. Each set is then used to generate the point midway between its members. For example, (5, 4, 4, 2) is one partition of 15 and would represent the point midway between 5, 4, 4 and 2 of a pentadecagon. Here is a graphical representation of p(30):

Here are graphical representations for the partitions 5 to 15, then 15 to 60 in increments of 5 (15, 20, 25, etc):

And here are some close-ups for the partitions of 35 and 40:

Post-Performative Post-Scriptum…

The title of this incendiary intervention wrily references the Bible:

The flowers appear on the earth; the time of the singing of birds is come, and the voice of the turtle is heard in our land — The Song of Solomon, 2:12

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:

M.i.P. Trip

Thursday, 8th Jan, 2015 by Krilling for Company in Fractals, Geometry, Mathematics and tagged animated gif, animated gifs, divide-and-discard, fractal, fractals, fractals based on squares, geometry, hexagon, hexagonal fractal, math, mathematics, maths, much in little, multum in parvo, recreational math, recreational mathematics, recreational maths, square, Squares | Leave a comment

The Latin phrase multum in parvo means “much in little”. It’s a good way of describing the construction of fractals, where the application of very simple rules can produce great complexity and beauty. For example, what could be simpler than dividing a square into smaller squares and discarding some of the smaller squares?

Yet repeated applications of divide-and-discard can produce complexity out of even a 2×2 square. Divide a square into four squares, discard one of the squares, then repeat with the smaller squares, like this:

Increase the sides of the square by a little and you increase the number of fractals by a lot. A 3×3 square yields these fractals:

And the 4×4 and 5×5 fractals yield more:

The Hex Fractor

Monday, 1st Dec, 2014 by Krilling for Company in Fractals, Geometry, Mathematics and tagged animated gif, animated gifs, equilateral triangle, fractal, fractals, geometry, hex-ring, hex-rings, hexagon, hexagonal fractal, math, mathematics, maths, polygon, Polygons, recreational math, recreational mathematics, recreational maths, regular polygons, triangle | Leave a comment

A regular hexagon can be divided into six equilateral triangles. An equilateral triangle can be divided into three more equilateral triangles and a regular hexagon. If you discard the three triangles and repeat, you create a fractal, like this:

Adjusting the sides of the internal hexagon creates new fractals:

Discarding a hexagon after each subdivision creates new shapes:

And you can start with another regular polygon, divide it into triangles, then proceed with the hexagons:

Spijit

Monday, 20th Oct, 2014 by Krilling for Company in Base Behavior, Mathematics, Ulam Spirals and tagged all-base function, animated gif, animated gifs, Arithmetic, base 2, bases, binary, binary numbers, count of 0s, count of 1s, count of ones, count of zeroes, count of zeros, different bases, digit spiral, digit spirals, digits, integer, integers, math, mathematics, maths, number spiral, one, ones, prime 1014719, recreational math, recreational mathematics, recreational maths, spijit, spijits, spiral, spiral digit, spiral digits, spirals, zero, Zeroes, zeros | Leave a comment

The only two digits found in all standard bases are 1 and 0. But they behave quite differently. Suppose you take the integers 1 to 100 and compare the number of 1s and 0s in the representation of each integer, n, in bases 2 to n-1. For example, 10 would look like this:

1010 in base 2
101 in base 3
22 in base 4
20 in base 5
14 in base 6
13 in base 7
12 in base 8
11 in base 9

So there are nine 1s and four 0s. If you check 1 to 100 using this all-base function, the count of 1s goes like this:

1, 1, 2, 3, 5, 5, 8, 5, 9, 9, 11, 10, 15, 12, 14, 13, 15, 12, 17, 14, 20, 19, 20, 15, 23, 19, 22, 22, 25, 24, 31, 21, 25, 24, 24, 27, 33, 27, 31, 29, 34, 29, 36, 30, 34, 35, 34, 30, 40, 33, 36, 35, 38, 34, 42, 37, 43, 40, 41, 37, 48, 39, 42, 42, 44, 43, 48, 43, 47, 46, 51, 42, 53, 44, 48, 50, 51, 50, 55, 48, 59, 55, 55, 54, 64, 57, 57, 55, 60, 57, 68, 60, 64, 63, 64, 59, 68, 58, 61, 63.

And the count of 0s goes like this:

0, 1, 0, 2, 1, 2, 0, 4, 4, 4, 2, 5, 1, 2, 2, 7, 4, 8, 4, 7, 4, 3, 1, 8, 4, 4, 6, 8, 4, 7, 1, 10, 8, 7, 7, 12, 5, 6, 5, 10, 4, 8, 2, 6, 7, 4, 2, 12, 6, 9, 7, 8, 4, 11, 6, 10, 5, 4, 2, 12, 2, 3, 5, 14, 11, 13, 7, 10, 8, 11, 5, 17, 7, 8, 10, 10, 8, 10, 4, 13, 12, 10, 8, 16, 8, 7, 7, 12, 6, 14, 6, 8, 5, 4, 4, 16, 6, 10, 11, 15.

The bigger the numbers get, the bigger the discrepancies get. Sometimes the discrepancy is dramatic. For example, suppose you represented the prime 1014719 in bases 2 to 1014718. How 0s would there be? And how many 1s? There are exactly nine zeroes:

1014719 = 11110111101110111111 in base 2 = 1220112221012 in base 3 = 40B27B in base 12 = 1509CE in base 15 = 10[670] in base 1007.

But there are 507723 ones. The same procedure applied to the next integer, 1014720, yields 126 zeroes and 507713 ones. However, there is a way to see that 1s and 0s in the all-base representation are behaving in a similar way. To do this, imagine listing the individual digits of n in bases 2 to n-1 (or just base 2, if n <= 3). When the digits aren’t individual they look like this:

1 = 1 in base 2
2 = 10 in base 2
3 = 11 in base 2
4 = 100 in base 2; 11 in base 3
5 = 101 in base 2; 12 in base 3; 11 in base 4
6 = 110 in base 2; 20 in base 3; 12 in base 4; 11 in base 5
7 = 111 in base 2; 21 in base 3; 13 in base 4; 12 in base 5; 11 in base 6
8 = 1000 in base 2; 22 in base 3; 20 in base 4; 13 in base 5; 12 in base 6; 11 in base 7
9 = 1001 in base 2; 100 in base 3; 21 in base 4; 14 in base 5; 13 in base 6; 12 in base 7; 11 in base 8
10 = 1010 in base 2; 101 in base 3; 22 in base 4; 20 in base 5; 14 in base 6; 13 in base 7; 12 in base 8; 11 in base 9

So the list would look like this:

1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 2, 0, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 0, 0, 0, 2, 2, 2, 0, 1, 3, 1, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0, 2, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 0, 1, 0, 1, 0, 1, 2, 2, 2, 0, 1, 4, 1, 3, 1, 2, 1, 1

Suppose that these digits are compared against the squares of a counter-clockwise spiral on a rectangular grid. If the spiral digit is equal to 1, the square is filled in; if the spijit is not equal to 1, the square is left blank. The 1-spiral looks like this:

Now try zero. If the spijit is equal to 0, the square is filled in; if not, the square is left blank. The 0-spiral looks like this:

And here’s an animated gif of the n-spiral for n = 0..9:

Priamonds and Pearls

Saturday, 22nd Mar, 2014 by Krilling for Company in Mathematics, Prime numbers and tagged animated gif, animated gifs, Arithmetic, integer, integers, math, mathematics, maths, pearl necklace, priamond, priamonds, prime lines, prime numbers, prime patterns, prime pearls, prime-diamonds, prime-ladder, prime-ladders, primes, recreational math, recreational mathematics, recreational maths | Leave a comment

Interesting patterns emerge when primes are represented as white blocks in a series of n-width left-right lines laid vertically, one atop the other. When the line is five blocks wide, the patterns look like this (the first green block is 1, followed by primes 2, 3 and 5, then 7 in the next line):

(Click for larger version)

Right at the bottom of the first column is an isolated prime diamond, or priamond (marked with a green block). It consists of the four primes 307-311-313-317, where the three latter primes equal 307 + 4 and 6 and 10, or 307 + 5-1, 5+1 and 5×2 (the last prime in the first column is 331 and the first prime in the second is 337). About a third of the way down the first column is a double priamond, consisting of 97, 101, 103, 107, 109 and 113. For a given n, then, a priamond is a set of primes, p₁, p₂, p₃ and p₄, such that p₂ = p₁ + n-1, p₃ = p + n+1 and p₄ = p₁ + 2n.

There are also fragments of pearl-necklace in the columns. One is above the isolated priamond. It consists of four prime-blocks slanting from left to right: 251-257-263-269, or 251 + 6, 12 and 18. A prearl-necklace, then, is a set of primes, p₁, p₂, p₃…, such that p₂ = p₁ + n+i, p₃ = p + 2(n+i)…, where i = +/-1. Now here are the 7-line and 9-line:

Above: 7-line for primes

Above: 9-line for primes

In the 9-line, you can see a prime-ladder marked with a red block. It consists of the primes 43-53-61-71-79-89-97-107, in alternate increments of 10 and 8, or 9+1 and 9-1. A prime-ladder, then, is a set of primes, p₁, p₂, p₃, p₄…, such that p₂ = p₁ + n+1, p₃ = p + 2n, p₃ = p + 3n+1…

And here is an animated gif of lines 5 through 51:

(Click or open in new window for larger version or if file fails to animate)

Hex Appeal

Tuesday, 7th Jan, 2014 by Krilling for Company in Geometry, Mathematics, Rep-Tiles and tagged animated gif, animated gifs, double-triangle, equilateral triangle, equilateral triangles, fish rep-tile, fish rep-tiles, five sides, five vertices, four right triangles, geometric, geometric dissection, geometric dissections, geometry, Great Sphinx at Giza, hexiamonds, isosceles right triangle, isosceles right triangles, math, mathematics, maths, pentagon, pentagonal rep-tile, pentagonal rep-tiles, Pentagons, polyiamond, polyiamonds, polyshape, polyshapes, recreational math, recreational mathematics, recreational maths, rep-4 sphinx, rep-9 sphinx, rep-tile, Rep-Tiles, right triangle, right triangles, self-replication, sphinx, sphinx hexiamond, tessellation, tessellations, three equilateral triangles, tiling, tilings, triangle, Triangles, vertex | Leave a comment

A polyiamond is a shape consisting of equilateral triangles joined edge-to-edge. There is one moniamond, consisting of one equilateral triangle, and one diamond, consisting of two. After that, there are one triamond, three tetriamonds, four pentiamonds and twelve hexiamonds. The most famous hexiamond is known as the sphinx, because it’s reminiscent of the Great Sphinx of Giza:

The Great Sphinx of Giza

It’s famous because it is the only known pentagonal rep-tile, or shape that can be divided completely into smaller copies of itself. You can divide a sphinx into either four copies of itself or nine copies, like this (please open images in a new window if they fail to animate):

So far, no other pentagonal rep-tile has been discovered. Unless you count this double-triangle as a pentagon:

It has five sides, five vertices and is divisible into sixteen copies of itself. But one of the vertices sits on one of the sides, so it’s not a normal pentagon. Some might argue that this vertex divides the side into two, making the shape a hexagon. I would appeal to these ancient definitions: a point is “that which has no part” and a line is “a length without breadth” (see Neuclid on the Block). The vertex is a partless point on the breadthless line of the side, which isn’t altered by it.

But, unlike the sphinx, the double-triangle has two internal areas, not one. It can be completely drawn with five continuous lines uniting five unique points, but it definitely isn’t a normal pentagon. Even less normal are two more rep-tiles that can be drawn with five continuous lines uniting five unique points: the fish that can be created from three equilateral triangles and the fish that can be created from four isosceles right triangles:

Rep It Up

Tuesday, 17th Dec, 2013 by Krilling for Company in Geometry, Mathematics, Rep-Tiles and tagged animated gifs, equilateral triangle, equilateral triangles, geometric, geometric dissection, geometric dissections, geometry, isosceles right triangle, isosceles right triangles, math, mathematics, maths, polyiamond, polyiamonds, polyshape, polyshapes, recreational math, recreational mathematics, recreational maths, rep-tile, Rep-Tiles, right triangle, right triangles, self-replication, tessellation, tessellations, tiling, tilings, triangle, Triangles, twelve equilateral triangles | Leave a comment

When I started to look at rep-tiles, or shapes that can be divided completely into smaller copies of themselves, I wanted to find some of my own. It turns out that it’s easy to automate a search for the simpler kinds, like those based on equilateral triangles and right triangles.

(Please open the following images in a new window if they fail to animate)

Previously pre-posted (please peruse):

• Rep-Tile Reflections

Hextra Texture

Thursday, 5th Dec, 2013 by Krilling for Company in Fractals, Geometry, Mathematics and tagged animated fractal, animated fractals, animated gif, animated gifs, equilateral triangle, equilateral triangles, fractal, fractal method, fractal methods, fractals, geometric, geometry, hexagon, Hexagons, math, mathematics, maths, recreational math, recreational mathematics, recreational maths, triangle, Triangles | Leave a comment

A hexagon can be divided into six equilateral triangles. An equilateral triangle can be divided into a hexagon and three more equilateral triangles. These simple rules, applied again and again, can be used to create fractals, or shapes that echo themselves on smaller and smaller scales.

Previously pre-posted (please peruse):

• Fractal Fourmulas

Fractal Fourmulas

Monday, 25th Nov, 2013 by Krilling for Company in Fractals, Geometry, Mathematics and tagged animated fractal, animated fractals, animated gif, animated gifs, fractal, fractal method, fractal methods, fractals, geometric, geometry, math, mathematics, maths, recreational math, recreational mathematics, recreational maths, right triangle, right triangles, square, Squares, triangle, Triangles | Leave a comment

A square can be divided into four right triangles. A right triangle can be divided into a square and two more right triangles. These simple rules, applied again and again, can be used to create fractals, or shapes that echo themselves on smaller and smaller scales.

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

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