Tag Archives: recreational math

Takes on Flakes

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

The Koch Snowflake is a fractal named after the Swedish mathematician Helge von Koch (1870-1924). It’s simple to make, attractive to see:

A Koch snowflake

And here’s how you making it, starting with an equilateral triangle:

Make the Flake #0

Make the Flake #1: Erect a 1/9-sized equilateral triangle on middle of each side

Make the Flake #2: Then a 1/81-sized equilateral triangle on the middle of each straight line created by #1

Make the Flake #3: And so on.

Make the Flake #4

Make the Flake #5

Make the Flake #6

Make the Flake #0-6 (animated at ezGif)

In the end, the Koch snowflake has an infinitely long perimeter around a finite area (see discussion at Wikipedia). The Koch anti-snowflake or un-flake also combines an infinitely long perimeter and finite area:

Koch un-Flake

You make the un-flake like this:

Make the Un-Flake #0

Make the Un-Flake #1

Make the Un-Flake #2

Make the Un-Flake #3

Make the Un-Flake #4

Make the Un-Flake #5

Make the Un-Flake #6

Make the Un-Flake #0-6 (animated at ezGif)

And you can combine the Koch snowflake and the Koch anti-snowflake like this:

Make the Flake+Un-Flake #0

Make the Flake+Un-Flake #1

Make the Flake+Un-Flake #2

Make the Flake+Un-Flake #3

Make the Flake+Un-Flake #4

Make the Flake+Un-Flake #5

Make the Flake+Un-Flake #6

Make the Flake+Un-Flake #0-6 (animated at ezGif)

Here’s another variation, what you might call the Koch-Cross flake:

Koch-Cross Flake #0

Koch-Cross Flake #1

Koch-Cross Flake #2

Koch-Cross Flake #3

Koch-Cross Flake #4

Koch-Cross Flake #5

Koch-Cross Flake #6

Koch-Cross Flake #0-6 (animated at ezGif)

And the Koch-Cross un-flake:

Koch-Cross un-Flake

And combined Koch-Cross Flake+un-Flake:

Combined Koch-Cross Flake+un-Flake

Here’s an animation of the combined Koch-Cross Flake+un-Flake:

Combined Koch-Cross Flake+un-Flake (animated at ezGif)

And the combined Koch-Cross Flake+un-Flake created on the sides of a square:

Combined Koch-Cross Flake+un-Flake on square

And a final variant of the infinitely many on offer:

Variant Koch snowflake stage #1

Variant Koch snowflake stage #6

Un-Flake of variant Koch snowflake

Combined variant Koch Flake+un-Flake

Back in Frac

by in Album Art, Arithmetic, Art, Fractions, Mathematics, Optical Illusions, Powers, Square Roots and tagged , , , , , , , , , , , , , , , | Leave a comment

Here’s a graph representing the fractional parts of √n for n = 1 to 1832, with frac(√1) at the top left and frac(√1832) at the bottom right:

graph for frac(√n), n = 1..1832

There’s an odd optical illusion making it seem as though each set of triangular waves ends lower on the right than it starts on the left. Otherwise, it’s a dull graph, because the fractional part of √n simply rises towards 0.9999…, then falls to 0 like this:

0 = frac(1) = frac(√1)
0.414213562373… = frac(1.414213562373…) = frac(√2)
0.732050807568… = frac(1.732050807568…) = frac(√3)
0 = frac(2) = frac(√4)
0.236067977499… = frac(2.236067977499…) = frac(√5)
0.449489742783… = frac(2.449489742783…) = frac(√6)
0.645751311064… = frac(2.645751311064…) = frac(√7)
0.828427124746… = frac(2.828427124746…) = frac(√8)
0 = frac(3) = frac(√9)
0.162277660168… = frac(3.162277660168…) = frac(√10)
0.316624790355… = frac(3.316624790355…) = frac(√11)
0.464101615137… = frac(3.464101615137…) = frac(√12)
0.605551275463… = frac(3.605551275463…) = frac(√13)
0.741657386773… = frac(3.741657386773…) = frac(√14)
0.872983346207… = frac(3.872983346207…) = frac(√15)
0 = frac(4) = frac(√16)
0.123105625617… = frac(4.123105625617…) = frac(√17)
0.242640687119… = frac(4.242640687119…) = frac(√18)
0.358898943540… = frac(4.358898943540…) = frac(√19)
0.472135954999… = frac(4.472135954999…) = frac(√20)

But what about the fractional parts of the sum of √n? What does that graph look like? Much more interesting:

frac(sum(√n)), n = 1..1832

Here are the fractional parts for the sum of √n:

0 = frac(1) = frac(sum(√1))
0.414213562373… = frac(02.414213562373…) = frac(sum(√1..√2))
0.146264369941… = frac(04.146264369941…) = frac(sum(√1..√3))
0.146264369941… = frac(06.146264369941…) = frac(sum(√1..√4))
0.382332347441… = frac(08.382332347441…) = frac(sum(√1..√5))
0.831822090224… = frac(10.831822090224…) = frac(sum(√1..√6))
0.477573401289… = frac(13.477573401289…) = frac(sum(√1..√7))
0.306000526035… = frac(16.306000526035…) = frac(sum(√1..√8))
0.306000526035… = frac(19.306000526035…) = frac(sum(√1..√9))
0.468278186204… = frac(22.468278186204…) = frac(sum(√1..√10))
0.784902976559… = frac(25.784902976559…) = frac(sum(√1..√11))
0.249004591697… = frac(29.249004591697…) = frac(sum(√1..√12))
0.854555867161… = frac(32.854555867161…) = frac(sum(√1..√13))
0.596213253935… = frac(36.596213253935…) = frac(sum(√1..√14))
0.469196600142… = frac(40.469196600142…) = frac(sum(√1..√15))
0.469196600142… = frac(44.469196600142…) = frac(sum(√1..√16))
0.592302225760… = frac(48.592302225760…) = frac(sum(√1..√17))
0.834942912879… = frac(52.834942912879…) = frac(sum(√1..√18))
0.193841856420… = frac(57.193841856420…) = frac(sum(√1..√19))
0.665977811419… = frac(61.665977811419…) = frac(sum(√1..√20))

As with square roots, so with cube roots. The graph of frac(∛n) looks like this:

frac(∛n), n = 1..1832

It’s dull again, because the fractional part of ∛n is simply rising towards 0.9999…, then falling to 0. Just more slowly. But the graph of frac(sum(∛n)) looks like this:

frac(sum(∛n))

But roots don’t end with √n and ∛n, of course. Those roots represent n^(1/2) and n^(1/3), respectively, because when x = n^(a/b), n = x^(b/a). What about the graph of sum(n^(4/5)), where n = (n^4/5)^(5/4)? The graph looks like this:

frac(sum(n^(4/5)))

One of the curves in that graph reminds of the cephalopodic tentacles in Jean Delville’s marvellous painting Les Trésors de Sathan (sic) (Treasures of Satan) (1895), which was used on the cover of Morbid Angel’s Blessed Are the Sick (1991). The curve is longer in frac(sum(n^(84/97))):

frac(sum(n^(84/97))) (curve in red)

Les Trésors de Sathan (1895) by Jean Delville as the cover of Blessed Are the Sick (1991)

frac(sum(n^(4/5))) (curve in red)

Those Delvillean curves are examples of how, as the a/b of n^(a/b) climbs from 1/b to (b-1)/b, the graph of frac(sum(n^(a/b))) changes in interesting ways. Here are the graphs for n^(1..10/11):

frac(sum(n^(1/11)))

frac(sum(n^(2/11)))

frac(sum(n^(3/11)))

frac(sum(n^(4/11)))

frac(sum(n^(5/11)))

frac(sum(n^(6/11)))

frac(sum(n^(7/11)))

frac(sum(n^(8/11)))

frac(sum(n^(9/11)))

frac(sum(n^(10/11)))

Here’s an animation of those graphs:

animation of frac(sum(n^(1/11..10/11))) (created at ezGif)

And here are graphs for n^(72/83) and n^(82/83), with a Delvillean curve in the graph of 72/83 and domes in the graph of 82/83:

frac(sum(n^(72/83)))

frac(sum(n^(82/83)))

As the denominators get bigger, so do the domes:

frac(sum(n^(1000/1009)))

frac(sum(n^(1001/1009)))

frac(sum(n^(1002/1009)))

frac(sum(n^(1003/1009)))

frac(sum(n^(1004/1009)))

frac(sum(n^(1005/1009)))

frac(sum(n^(1006/1009)))

frac(sum(n^(1007/1009)))

frac(sum(n^(1008/1009)))

Here’s an animation of those graphs:

animation of frac(sum(n^(1000/1009..1008/1009))) (created at ezGif)

Previously Pre-Posted…

• Think Frinc — an earlier look at fraction-patterns
• Altars of Mathness — and similar patterns from integer-digits

Elsewhere Other-Accessible…

• Jean Delville at Wikipedia
• Blessed Are the Sick at BandCamp

Fungible Fractals

by in Circles, Fractals, Geometry, Mathematics, Polygons, Sierpiński triangle and tagged , , , , , | Leave a comment

Thinking it over, I’ve decided that trircle is a much better name than ciangle:

A Sierpiński triangle

↓

A Sierpiński trircle, or triangle converted into a circle

A trircle is a triangle converted into a circle; a ciangle is a circle converted into a triangle. Or another regular polygon. The trircle reminded me that circularized fractals are fungible, because the Sierpiński triangle can be converted into another regular polygon like a square or pentagon or hexagon. You can go viâ the trircle, but you don’t have to. The point is that the Sierpiński triangle has a center and points lying at some distance and some angle 0° through 360°, so you can easily adjust the points to fit inside any other regular polygon:

A Sierpiński triangle again

↓

A Sierpiński trare, or triangle converted into square

↓

A Sierpiński trentagon

↓

A Sierpiński trexagon

↓

A Sierpiński treptagon

↓

A Sierpiński troctogon

Here’s an animation of the conversions:

Sierpiński triangle → square, pentagon, hexagon, heptagon, octagon (animated at ezGif)

A Sierpiński carpet, with points lying at 0° through 360° inside a square, is similarly fungible:

A Sierpiński carpet

↓

A Sierpiński carpet converted into a circle

↓

A Sierpiński carpet converted into a triangle

↓

A Sierpiński carpet again

↓

A Sierpiński carpet converted into a pentagon

↓

↓

↓

And an animation of the carpet conversions:

Sierpiński carpet → triangle, pentagon, hexagon, heptagon, octagon (animated at ezGif)

Finally, and fungibly, the fractal that I call the centered Sierpiński triangle:

A centered Sierpiński triangle

↓

A centered Sierpiński trircle, or triangle converted into a circle

↓

↓

↓

↓

↓

And the final animation:

Centered Sierpiński triangle → square, pentagon, hexagon, heptagon, octagon (animated at ezGif)

Flowly We Rote

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

As the old mathematical joke goes: A topologist is someone who can’t tell the difference between a coffee-cup and a donut. That’s because topology is, crudely speaking, the branch of geometry that studies shapes when the distance and angle between one part and another doesn’t matter. For example, how can (or can’t) shapes flow smoothly into each other, without being cut or torn or pierced? The shape of a perfectly plastic substance can flow smoothly from that of a coffee-cup into that of a donut. And vice versa:

Topologically speaking, a coffee-cup is the same as a donut (Wikipedia)

That’s topology in three dimensions. I came across some unexpected topology in two dimensions when I was looking at transformations of a triangle — the Sierpiński triangle, a fractal named after the Polish mathematician Wacław Sierpiński (1882-1969):

Sierpiński triangle

I wondered what happened when you rotate the points inside a Sierpiński triangle while the triangular boundary remains fixed. That is, each point stays at the same position in the width between the center and the boundary as the whole interior flows around the center of the triangle:

Points inside a Sierpiński triangle rotated by 1°

Points inside a Sierpiński triangle rotated by 2°

Points inside a Sierpiński triangle rotated by 3°

Points rotated by 4°

Points rotated by 5°

Points rotated by 10°

Points rotated by 20°

Points rotated by 30°

Points rotated by 40°

Points rotated by 50°

Points rotated by 60°

Points rotated by 70°

Points rotated by 80°

Points rotated by 90°

WARNING! If you’re sensitive to flickering images, please note that there are flickering animated gifs below

Here’s the flowing rotation from 0° to 90° animated in a gif:

Interior points of Sierpiński triangle flowing 0° → 90° around center (animated at ezGif)

And here’s the whole rotating flow from 0° to 120° (which maps the points back onto themselves):

Interior points of Sierpiński triangle flowing continuously around center (slow animation)

Interior points flowing continuously around the center (faster animation)

As you can see, the appearance of the Sierpiński triangle changes notably as the points rotate: the rotations aren’t rotationally symmetrical (in the standard sense). Sometimes a rotation looks like a stumpy triskelion, a three-legged shape like the flag of the Isle of Man:

Triskelion on the Manx flag (Wikipedia)

Interior points of a Sierpiński triangle rotated by 30°

But topologically speaking, each rotated triangle is the same (just as, topologically speaking, a coffee-cup is the same as a donut). You can see how they’re topologically the same by imagining that the triangle is stretched into a circle, like this:

Sierpiński triangle

↓

Sierpiński triangle stretched into circle

When you circularize the rotated triangles, all the circularized triangles are rotationally symmetrical:

Points rotated by 30°

↓

Circle from triangle rotated by 30°

Points rotated by 60°

↓

Circle from triangle rotated by 60°

Points rotated by 90°

↓

Circle from triangle rotated by 90°

Here are two animated gifs of the circularized triangles rotating:

Circularized Sierpiński triangle flowing around center (slow animation at ezGif)

Circularized Sierpiński triangle flowing around center (faster animation)

Here’s what I call the centered Sierpiński triangle turned into a circle:

Centered Sierpiński triangle

↓

Circle from centered Sierpiński triangle

And finally, the circularized centered Sierpiński triangle flowly rotating at two speeds:

Circularized centered Sierpiński triangle flowing around center (slow animation at ezGif)

Circularized centered Sierpiński triangle flowing around center (faster animation)

Tie-Phi, Cy-Phi

by in Arithmetic, √2, Continued Fractions, Fibonacci Sequence, Fractions, Integer Sequences, Irrational numbers, φ, Mathematics, Phi, Square Numbers, Square root of 2, Square Roots and tagged , , , , , , | Leave a comment

More and more slowly. That’s how this function increases:

x = x + 1/x

x = 1
1 + 1/1 = 2
2 + 1/2 = 2_1/2
2_1/2 + 1/2_1/2 = 2_9/10
2_9/10 + 1/2_9/10 = 3_71/290
3_71/290 + 1/3_71/290 = 3_150911/272890 = 3.5530103704…
3.5530103704… + 1/3.5530103704… = 3.8344618428…
3.8344618428… + 1/3.8344618428… = 4.0952546322…
4.0952546322… + 1/4.0952546322… = 4.33943969272…
4.3394396927… + 1/4.3394396927… = 4.56988419035…
4.5698841903… + 1/4.5698841903… = 4.78870811637…
4.7887081163… + 1/4.7887081163… = 4.99753270449…
4.9975327044… + 1/4.9975327044… = 5.19763144503…
[…]

But you can tie the function down, as it were, by changing it to this:

x = 1 + 1/x

x = 1
1 + 1/1 = 2
1 + 1/2 = 1_1/2 = 1.5
1 + 1/1_1/2 = 1_2/3 = 1.66666…
1 + 1/1_2/3 = 1_3/5 = 1.6
1 + 1/1_3/5 = 1_5/8 = 1.625
1 + 1/1_5/8 = 1_8/13 = 1.615384615384615384…
1 + 1/1_8/13 = 1_13/21 = 1.619047619047619047619047619…
1 + 1/1_13/21 = 1_21/34 = 1.617647058823529411764705882…
1 + 1/1_21/34 = 1_34/55 = 1.6181818…
1 + 1/1_34/55 = 1_55/89 = 1.617977528089887640449438202…
1 + 1/1_55/89 = 1_89/144 = 1.6180555…
1 + 1/1_89/144 = 1_144/233 = 1.618025751072961373390557940…
1 + 1/1_144/233 = 1_233/377 = 1.618037135278514588859416446…
1 + 1/1_233/377 = 1_377/610 = 1.618032786885245901639344262…
1 + 1/1_377/610 = 1_610/987 = 1.618034447821681864235055724…
1 + 1/1_610/987 = 1_987/1597 = 1.618033813400125234815278648…
1 + 1/1_987/1597 = 1_1597/2584 = 1.618034055727554179566563468…
1 + 1/1_1597/2584 = 1_2584/4181 = 1.618033963166706529538387946…
1 + 1/1_2584/4181 = 1_4181/6765 = 1.618033998521803399852180340…
1 + 1/1_4181/6765 = 1_6765/10946 = 1.618033985017357938973140873…
[…]

You could call the function “x = 1 + 1/x” a tie-phi, because it generates the golden ratio, a fascinating mathematical constant also known as phi or φ = 1.6180339887498948482… Note how the denominator (the lower part) of each fraction above becomes the numerator (the upper part) of the next fraction. The function is reproducing the Fibonacci sequence, which starts with “1, 1” and proceeds by adding the two previous numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597…

Dividing each number by the previous number yields a better and better approximation of φ (but never reaches the exact value of φ, which is an irrational number, that is, it cannot be expressed as a ratio of two whole numbers). And what about the function x = 1 – 1/x? If you remember that subtracting a negative number is the same as adding a positive number, you can work it out like this:

x = 1 – 1/x

x = 2

1 – 1/2 = 1/2 = 0.5
1 – 1/1/2 = -1
1 – 1/-1 = 2
1 – 1/2 = 1/2 = 0.5
1 – 1/1/2 = -1
1 – 1/-1 = 2
1 – 1/2 = 1/2 = 0.5
1 – 1/1/2 = -1
1 – 1/-1 = 2
[…]

x = 3

1 – 1/3 = 2/3 = 0.66666…
1 – 1/2/3 = -1/2 = -0.5
1 – 1/-1/2 = 3
1 – 1/3 = 2/3 = 0.66666…
1 – 1/2/3 = -1/2 = -0.5
1 – 1/-1/2 = 3
1 – 1/3 = 2/3 = 0.66666…
1 – 1/2/3 = -1/2 = -0.5
1 – 1/-1/2 = 3 = 3
1 – 1/3 = 2/3 = 0.66666…
[…]

x = 73

1 – 1/73 = 72/73 = 0.98630136986301369863013…
1 – 1/72/73 = -1/72 = -0.01388888…
1 – 1/-1/72 = 73
1 – 1/73 = 72/73 = 0.98630136986301369863013…
1 – 1/72/73 = -1/72 = -0.01388888…
1 – 1/-1/72 = 73
1 – 1/73 = 72/73 = 0.98630136986301369863013…
1 – 1/72/73 = -1/72 = -0.01388888…
1 – 1/-1/72 = 73
[…]

The function falls into an endless loop of period 3. But what about using the two functions together, first x = 1 + 1/x, then x = 1 – 1/x, then repeating?

loop(x = 1 + 1/x, x = 1 – 1/x)

x = 1
1 + 1/1 = 2
1 – 1/2 = 1/2 = 0.5
1 + 1/1/2 = 3
1 – 1/3 = 2/3 = 0.666666…
1 + 1/2/3 = 2_1/2 = 2.5
1 – 1/2_1/2 = 3/5 = 0.6
1 + 1/3/5 = 2_2/3 = 2.66666…
1 – 1/2_2/3 = 5/8 = 0.625
1 + 1/5/8 = 2_3/5 = 2.6
1 – 1/2_3/5 = 8/13 = 0.615384615384…
1 + 1/8/13 = 2_5/8 = 2.625
1 – 1/2_5/8 = 13/21 = 0.619047619047619…
1 + 1/13/21 = 2_8/13 = 2.615384615384615384…
1 – 1/2_8/13 = 21/34 = 0.6176470588235294117647058824…
1 + 1/21/34 = 2_13/21 = 2.619047619047619047619047619…
1 – 1/2_13/21 = 34/55 = 0.6181818181818181818181818182…
1 + 1/34/55 = 2_21/34 = 2.617647058823529411764705882…
1 – 1/2_21/34 = 55/89 = 0.6179775280898876404494382022…
1 + 1/55/89 = 2_34/55 = 2.6181818…
1 – 1/2_34/55 = 89/144 = 0.618055555…
1 + 1/89/144 = 2_55/89 = 2.617977528089887640449438202…
1 – 1/2_55/89 = 144/233 = 0.6180257510729613733905579399…
1 + 1/144/233 = 2_89/144 = 2.618055555…
1 – 1/2_89/144 = 233/377 = 0.6180371352785145888594164456…
1 + 1/233/377 = 2_144/233 = 2.618025751072961373390557940…
1 – 1/2_144/233 = 377/610 = 0.6180327868852459016393442623…
1 + 1/377/610 = 2_233/377 = 2.618037135278514588859416446…
1 – 1/2_233/377 = 610/987 = 0.6180344478216818642350557244…
1 + 1/610/987 = 2_377/610 = 2.618032786885245901639344262…
1 – 1/2_377/610 = 987/1597 = 0.6180338134001252348152786475…

Phi and the Fibonacci sequence are back, but in a zig-zagging or two-steps-forward, one-step-back kind of way. Now adapt the two previous functions slightly. Here’s the plus function adapted:

x = 2 + 1/x

x = 1

2 + 1/1 = 3…
2 + 1/3 = 2_1/3 = 2.33333…
2 + 1/2_1/3 = 2_3/7 = 2.428571428571428571…
2 + 1/2_3/7 = 2_7/17 = 2.411764705882352941176470588…
2 + 1/2_7/17 = 2_17/41 = 2.414634146341463…
2 + 1/2_17/41 = 2_41/99 = 2.41414141…
2 + 1/2_41/99 = 2_99/239 = 2.414225941422594142259414226…
2 + 1/2_99/239 = 2_239/577 = 2.414211438474870017331022530…
2 + 1/2_239/577 = 2_577/1393 = 2.414213926776740847092605887…
2 + 1/2_577/1393 = 2_1393/3363 = 2.414213499851323223312518585…
[…]

This function is generating Pell numbers and approximating the value 1 + √2, where √2 = 1.414213562373095048801688724… Now try adapting the minus function:

x = 2 – 1/x

x = 2

2 – 1/2 = 1_1/2 = 1.5
2 – 1/1_1/2 = 1_1/3 = 1.33333…
2 – 1/1_1/3 = 1_1/4 = 1.25
2 – 1/1_1/4 = 1_1/5 = 1.2
2 – 1/1_1/5 = 1_1/6 = 1.166666…
2 – 1/1_1/6 = 1_1/7 = 1.142857142857142857…
2 – 1/1_1/7 = 1_1/8 = 1.125
2 – 1/1_1/8 = 1_1/9 = 1.11111…
2 – 1/1_1/9 = 1_1/10 = 1.1
2 – 1/1_1/10 = 1_1/11 = 1.09090909…
2 – 1/1_1/11 = 1_1/12 = 1.0833333…
2 – 1/1_1/12 = 1_1/13 = 1.0769230769230769230…
2 – 1/1_1/13 = 1_1/14 = 1.0714285714285714285…
2 – 1/1_1/14 = 1_1/15 = 1.0666666…
2 – 1/1_1/15 = 1_1/16 = 1.0625
[…]

Where x = 1 – 1/x falls into an endless loop, x = 2 – 1/x endlessly falls towards 1. Now try combining x = 2 + 1/x and x = 2 – 1/x. The result might be surprising:

loop(x = 2 + 1/x, x = 2 – 1/x)

x = 1

2 + 1/1 = 3
2 – 1/3 = 1_2/3 = 1.66666…
2 + 1/1_2/3 = 2_3/5 = 2.6
2 – 1/2_3/5 = 1_8/13 = 1.615384615384615384…
2 + 1/1_8/13 = 2_13/21 = 2.619047619047619047619047619…
2 – 1/2_13/21 = 1_34/55 = 1.618181818181818181818181818…
2 + 1/1_34/55 = 2_55/89 = 2.617977528089887640449438202…
2 – 1/2_55/89 = 1_144/233 = 1.618025751072961373390557940…
2 + 1/1_144/233 = 2_233/377 = 2.618037135278514588859416446…
2 – 1/2_233/377 = 1_610/987 = 1.618034447821681864235055724…
2 + 1/1_610/987 = 2_987/1597 = 2.618033813400125234815278648…
2 – 1/2_987/1597 = 1_2584/4181 = 1.618033963166706529538387946…
2 + 1/1_2584/4181 = 2_4181/6765 = 2.618033998521803399852180340…
2 – 1/2_4181/6765 = 1_10946/17711 = 1.618033990175597086556377393…
2 + 1/1_10946/17711 = 2_17711/28657 = 2.618033988205325051470844820…
2 – 1/2_17711/28657 = 1_46368/75025 = 1.618033988670443185604798401…
2 + 1/1_46368/75025 = 2_75025/121393 = 2.618033988780242682856507377…
2 – 1/2_75025/121393 = 1_196418/317811 = 1.618033988754322537608830406…
2 + 1/1_196418/317811 = 2_317811/514229 = 2.618033988748203621343798191…
2 – 1/2_317811/514229 = 1_832040/1346269 = 1.618033988749648101530971893…
2 + 1/1_832040/1346269 = 2_1346269/2178309 = 2.618033988749989097047296779…
2 – 1/2_1346269/2178309 = 1_3524578/5702887 = 1.618033988749908598925421458…
[…]

It’s a tie-phi again, with better and better aproximations to φ and φ + 1 = φ^2. Now another function that may be another surprise:

x = 5 – 5/x

x = 2

5 – 5/2 = 2_1/2 = 2.5
5 – 5/2_1/2 = 3 = 3
5 – 5/3 = 3_1/3 = 3.333333…
5 – 5/3_1/3 = 3_1/2 = 3.5
5 – 5/3_1/2 = 3_4/7 = 3.571428571428571428…
5 – 5/3_4/7 = 3_3/5 = 3.6
5 – 5/3_3/5 = 3_11/18 = 3.611111…
5 – 5/3_11/18 = 3_8/13 = 3.615384615384615384615384615…
5 – 5/3_8/13 = 3_29/47 = 3.61702127659574468085106383…
5 – 5/3_29/47 = 3_21/34 = 3.617647058823529411764705882…
5 – 5/3_21/34 = 3_76/123 = 3.617886178861788617886178862…
5 – 5/3_76/123 = 3_55/89 = 3.617977528089887640449438202…
5 – 5/3_55/89 = 3_199/322 = 3.618012422360248447204968944…
5 – 5/3_199/322 = 3_144/233 = 3.61802575107296137339055794…
5 – 5/3_144/233 = 3_521/843 = 3.618030842230130486358244365…
5 – 5/3_521/843 = 3_377/610 = 3.618032786885245901639344262…
5 – 5/3_377/610 = 3_1364/2207 = 3.618033529678296329859537834…
[…]

It’s a tie-phi with the Fibonacci sequence again. But only the Fibonacci sequence. The Lucas sequence too:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778…

The Lucas sequence is generated by the same rule as the Fibonacci sequence — add the two previous numbers — but it’s seeded with “2, 1” rather than “1, 1”.

Finally, if you’re wondering where cy-phi is, it’s here:

x = 1 – 1/x

x = φ

φ – 1/φ = 2 – φ = 0.3819660112501052…
2-φ – 1/(2-φ) = -φ = -1.6180339887498948…
-φ – 1/-φ = φ = 1.6180339887498948…
2-φ = 0.3819660112501052…
-φ = -1.6180339887498948…
φ 1.6180339887498948…
2-φ = 0.3819660112501052…
-φ = -1.6180339887498948…
[…]

That’s a cycling phi or cy-phi.

Hopping in Boundland

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

Shopping in Poundland. That’s what Brits do when they want cheap fills. Hopping in Boundland. That’s what rec-mathers do when they want deep thrills. As I’ve described previously, a point can create interesting geometry by jumping towards fixed vertices inside a polygon. But it can also create interesting geometry by jumping at fixed angles inside a polygon. Suppose a point jumps at 0°, 120° or 240° halfway towards the perimeter of a triangle. It’s bounded in its jumps, so you could say it’s hopping in Boundland. Or bounding in Hopland. Either way, it creates this pattern inside the polygon:

Jumping halfway towards perimeter of triangle at 0°, 120° or 240°

If you stretch the triangle-and-pattern into a circle, you get this (the graphics aren’t as good as they could be, but I’m working on it):

Triangle → circle, 1/2 at 0°, 120° or 240°

You can also introduce restrictions, like banning the point from jumping at the same angle twice in a row:

Ban on same angle twice in a row, 1/2 jump at 0°, 120° or 240°

Triangle → circle, ban j+0, 1/2 jump at 0°, 120° or 240°

If the jump is 2/3rds of the distance to the perimeter and the ban is on the same angle twice, the point generates this pattern by hopping in Boundland:

Triangle, 2/3 jump, ban j+0, 0°, 120° or 240°

Triangle → circle, 2/3 jump, ban j+0, 0°, 120° or 240°

Now try adjusting the angles. Swinging the angles 60° with the same 2/3rd jump and same ban, the point generates this:

Triangle, 2/3 jump, ban j+0, 60°, 180° or 300°

Triangle → circle, 2/3 jump, ban j+0, 60°, 180° or 300°

A point jumping 2/3rds at four angles inside a square generates this pattern:

Square, 2/3 jump, ban j+0, 45°, 135°, 225, 315°

Square → circle, 2/3 jump, ban j+0, 45°, 135°, 225, 315°

And what if the point jumps 5/7ths towards the center of the polygon, not just towards the perimeter? Then it generates this pattern inside a hexagon with a ban on jumping towards the angle adjacent clockwise to the angle it’s just jumped at:

Hexagon, 5/7 jump, ban j+1, 6 angles + center

The point can also jump inside a circle rather than a polygon. Here’s the pattern generating by a point jumping 2/3rds at three angles with a ban on the same jump twice in a row:

Circle, 2/3 jump, ban j+0, 3 angles

And if the point can jump towards the center of the circle too, it generates this pattern:

Circle, 2/3 jump, ban j+0, 3 angles + center

Elsewhere Other-Accessible

• Controlled Chaos — a look at points jumping towards vertices, not perimeters

Think Frink

by in Arithmetic, Fractions, Geometry, Mathematics, Reciprocals, Square Roots, Symmetry and tagged , , , , , , | Leave a comment

Inky Bloaters (1987) is the name of an album by psycho-songstress Danielle Dax. Frinky growthers are those who are interested in frincremental growth. That’s growth by fractions, as in the equation x = x + 1/x. If the initial x = 1, its frincremental growth looks like this:

1
1 + 1/1 = 1 + 1 = 2
2 + 1/2 = 2_1/2
2_1/2 + 1/2_1/2 = 2_9/10
2_9/10 + 1/2_9/10 = 3_71/290
3_71/290 + 1/3_71/290 = 3_150911/272890
3_150911/272890 → 3_220789390391/264588959090 → 4_25570190327910692085061/268440386798659418988490 → 4_100170363026578204006507990129853967021051645381/295105036840595214385430531020664149472669868290

As you can see, the numerators and denominators of the fractional part of x get very large very quickly. So you can’t track the frincremental growth of x with perfect accuracy. Even the most compendious computer will run out of space. But representing x as a decimal is usually enough for us frinky growthers:

1 + 1 = 2
2 + 1/2 = 2 + 0.5 = 2.5
2.5 + 1/2.5 = 2.9
3.244827586206896551724137931…
3.553010370478947561288431236…
3.834461842815967366750790750…
4.095254632258778985771918456…
4.339439692724345181049239663…
4.569884190357676650018985962…
4.788708116379690742064597208…
4.997532704493448986664559639…
5.197631445038131469095668466…
5.390026771750770995914851381…
5.575554607204394029915651664…
5.754908962142979073283550015…
5.928673657045750549124213874…
6.097345447373015508408978797…
6.261351244425377152997703626…
6.421061179383957004641284553…
6.576798676981813718180627345…

The larger x gets, the slower it grows. But it never stops growing and will pass any finite integer in finite time. I was interested in the fractional part of x += 1/x (a shorthand for x = x + 1/x). Plainly, it’s almost unique for every integer seed (the fractional part is identical, just displaced by one step, for initial x = 1 = 2). I graphed the fractional part of x += 1/x for 1, 2, 3, 4, 5… and discovered some interesting patterns:

frac(x) of x += 1/x for x = 1, 2, 3, 4, 5…

Later frac(x) of x += 1/x

In time, the rounded patterns disappear for lower initial x. But you also get symmetrical patterns for x += sqrt(x), that is, x = x + square_root(x). And they last longer:

frac(x) of x += sqrt(x) for 1, 2, 3, 4, 5… #1

frac(x) of x += sqrt(x) for 1, 2, 3, 4, 5… #2

frac(x) of x += sqrt(x) for 1, 2, 3, 4, 5… #3

frac(x) of x += sqrt(x) for 1, 2, 3, 4, 5… #4

frac(x) of x += sqrt(x) for 1, 2, 3, 4, 5… #5

Here’s an animation of the first fifty steps of x += sqrt(x):

animated frac(x) of x += sqrt(x) for 1, 2, 3, 4, 5… (animated at EZgif) (click for larger image)

And what about x += ln(x), or x = x + natural_logarithm(x)? The patterns are both more dynamic and longer-lasting for lower initial x:

frac(x) of x += ln(x) for 1, 2, 3, 4, 5… #1

frac(x) of x += ln(x) for 1, 2, 3, 4, 5… #2

frac(x) of x += ln(x) for 1, 2, 3, 4, 5… #3

frac(x) of x += ln(x) for 1, 2, 3, 4, 5… #4

And two animations of x += ln(x), one slower, one faster:

animated frac(x) of x += sqrt(x) for 1, 2, 3, 4, 5… (EZgif) (click for larger)

faster animated frac(x) of x += sqrt(x) for 1, 2, 3, 4, 5… (click for larger)

Middlemoth

by in Arithmetic, Base Behavior, Circles, Fractions, Geometry, Mathematics, Squares and tagged , , , , , , , , , , , , , | Leave a comment

I’ve never read Middlemarch (1871). But I have seen a middlemoth. It was when I was looking at a new way of creating fractLs. A fractL is what I call a graph shaped like a capital L, with the x- and y-axes representing values between 0 and 1, like 1/2 and 1/3 and 8/55. You can also use numbers > 1 to create numbers < 1: 73 → 0.73; 128719 → 0.128719; and so on. But I decided to reverse the integer before converting it: 73 → 0.37; 128719 → 0.917821; and so on. And use different bases for the x- and y-axes. So that’s what I did on a fractL: I mapped fractions converted from integers in one base against fractions converted from integers in another base. The results, as you can see, were spectacularly dull:

fractL for int→frac in base 2 and base 6

fractL for int→frac in base 3 and base 6

fractL b04, b06

fractL b06, b08

So I decided to try some perspectivision, mapping the integer-fractions not on a fractL but on a fractO instead. A fractO is a circle where you find a point inside the circle by using two fractions, fr1 and fr2, to create two radian values: θ1 = fr1 * 2 * π and θ2 = fr2 * 2 * π. Then you use θ1 and θ2 to find two points on the perimeter of the circle, (x1, y1) and (x2, y2), and then find their midpoint, (x3, y3) = ((x1, y1) + (x2, y2)) / 2. The results this time are much more pleasing on the eye:

fractO for integers in base 2 and base 6

↑

fractL b02, b06, for fractO b02, b06

Here’s an animated gif showing the conversion from visually dull fractL to visually interesting fractO:

fractL b02b06 to fractO b02b06 (animated at EZgif)

When I was looking at more fractOs, I found one that was lepidopterally interesting too:

fractO b09, b12 with middlemoth

fractO b09, b12 (middlemoth in green)

You can try spotting more pareidolia in more fractOs from reversed fractintegers:

fractO b03, b15

↑

fractL b03, b15 for fractO above

fractL b03b15 to fractO b03b15 (animated at EZgif)

fractO b02, b10

fractO b02, b12

fractO b02, b14

fractO b03, b06

fractO b03, b12

fractO b03, b21

fractO b04, b06

fractO b04, b20

fractO b06, b08

fractO b09, b15

fractO b10, b24

fractO b12, b16

fractO b15, b20

fractO b24, b28

fractO b42, b78

fractO b02, b18 (fr2 x 3)

fractO b02, b06 (fr2 x 3)

Altars of Mathness

by in Album Art, Arithmetic, Base Behavior, Fibonacci Sequence, Fractals, Integer Sequences, Mathematics, Polygonal numbers, Powers, Prime numbers, Triangular Numbers and tagged , , , , , , , , , , , | Leave a comment

What could be duller than digits? They just sit there on the page or screen, mindlessly marking mathematics:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100…

But perhaps they become more interesting as images. Let’s display the final digit of the integers, or counting numbers, on a graph. Running left-right and up-down, the graph represents the final or rightmost digit of 1, 2, 3, … 10, 11, 12, 13, … 100, 101, 102, 103, …, 1000, 1001, 1002, 1003, …:

Rightmost single digit of the integers (click for larger)

No, that’s still dull: the graph just generates endlessly repeating triangles. After all, the final digits fall into a cycle: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3… So do the final two digits: 1, 2, 3, 4, 5, […] 94, 95, 96, 97, 98, 99, 00, 01, 02, 03… Here they are as a graph:

Rightmost two digits of the integers

Now the triangles look like waves sweeping to shore. That’s a bit more interesting, but not much. So let’s try something different. The trailing digits of the integers generate triangles, so let’s see what the triangular numbers generate. The triangular numbers — 0, 1, 3, 6, 10, 15, 21… — are very simple to form. You just sum the integers: 1, 3 = 1 + 2, 6 = 1 + 2 + 3, 10 = 1 + 2 + 3 + 4, 15 = 1 + 2 + 3 + 4 + 5, 21 = 1 + 2 + 3 + 4 + 5 + 6, 28 = 1 + 2 + 3 + 4 + 5 + 6 + 7, 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8, 45 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9, 55 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10… Here are the final digits of the triangulars — 1, 3, 6, 0, 5, 1, 8, 6… — as a graph:

Final digit of triangular numbers in base 10 (click for larger)

Now something interesting has appeared. The final digits form a repeated palindromic pattern (counting 0 as the zero-th triangular number):

0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, 0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0, …

An Altar of Mathness created by the final digit of triangular numbers in base 10

And those palindromic digits create symmetric shapes that remind me of little altars — let’s call them “altars of mathness” in tribute to Morbid Angel’s genre-defining album Altars of Madness (1989). And what about the final two digits of the triangular numbers? Here’s the graph (adjusted so that 99 fits into the same space as 9):

Final two digits of triangulars in b10

Final two triangular digits in b10 (horizontal scale compressed)

The final two digits form palindromes too. And this time we don’t get just triangles, but curves too. But that’s in base 10. What happens with the trailing triangular digits in other bases? Well, here’s the final triangular digit creating more altars of mathness in different bases (note that the altars are more elaborate in even bases):

Final triangular digit in base 4

Final triangular digit in b5

Final triangular digit in b6

Final triangular digit in b7

Final triangular digit in b8

Final triangular digit in b9

Final triangular digit in b14

And here’s the graph for the final triangular digit in base 100:

Final triangular digit in b100

The graph for final single digit in b100 should look familiar, because it’s identical to the graph for final double triangular digits in b10:

Final two digits of triangulars in b10

That’s because two digits in b10 are equivalent in one digit in b100, four digits in b10 are equivalent to two digits in b100, and so on. But b100 can’t capture three digits in b10 (the graph is again adjusted so that 999 fits into the same space as 9 and 99 above):

Final three triangular digits in b10

If you compress the x-axis for that graph, you can see how long the symmetries are:

Final three triangular digits in b10 (x-axis / 2)

Final three triangular digits in b10 (x-axis / 4)

The final four digits of the triangulars in b10 create even longer symmetries:

Final quadruple triangular digits in b10

Final quadruple triangular digits in b10 (x-axis / 2)

Final quadruple triangular digits in b10 (x-axis / 8)

Note how, as the length of the final digits rises, you need to compress the x-axis more and more to see the symmetries. But integer sequences obviously don’t end with the counting numbers and triangulars. What about squares and powers of n? What about primes and Fibonacci numbers? Here’s the final two digits of the squares — 1, 4, 9, 16, 25, 49, 64, 81, 100, 121, 144, 169… — in b10:

Final two digits of the squares in b10

It’s reminiscent of the triangular numbers (so are the final-digit graphs for other polygonal numbers). So what about the powers of 2? That’s 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024… Here’s the graph for final single digits of 2^p in b10:

Final single digits of 2^p in b10

This time there’s repetition, but not symmetry. Here’s the graph for final double digits, or 2-digits, of 2^p in b10:

Final 2-dig of 2^p in b10

Now the graph looks a little like a range of eroded mountains. Now try dig-4, the final four digits of 2^p in b10:

Final 4-dig of 2^p in b10

The patterns are similar to those of dig-2 and don’t need compressing in the x-axis. This similarity and lack of need for compression are true of any number of final digits in 2^p. The final 10 digits look like this:

Final 10-dig of 2^p in b10

And the final 20 and 30 digits like this:

Final 20-dig of 2^p in b10

Final 30-dig of 2^p in b10

Powers don’t behave like polygonals: the finals are fractals. That is, the final digits create similar patterns at all scales: 1-dig, 2-dig, 10-dig, 100-dig, 1000-dig and so on. That’s true in other bases:

Final 5-dig of 3^p in b2

But a glimpse of b2 is all you’re going to get of other bases. There are other fish to fry — Fibonacci fish. The Fibonacci sequence, whose terms are equal to the sum of the previous two numbers (after seeding with “1, 1”), starts like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811… And what about the graphs for final fib-digits? As you’ll see, final Fib-digits are fractal too. Indeed, Fibonacci final-graphs look like 2-power final-graphs (in a way, Fibonacci numbers are powers of φ = 1.6180339887498948482…). The patterns are similar at all scales. And they remind me of the skyline of a ruined city in an Oriental tale, with collapsed domes and crumbling minarets:

Final 1-dig of Fibonacci numbers in b10

Final 2-fibdig in b10

Final 3-fibdig in b10

Final 4-fibdig in b10

Final 5-fibdig in b10

Final 10-fibdig in b10

Final 15-fibdig in b10

Final 20-fibdig in b10

Final 25-fibdig in b10

So final fibdigs are fractal. But final prime digits aren’t:

Final 1-digit of primes in b10

Final 1-digit of primes in b5

Final 2-digit of primes in b10

Primes aren’t final-digitally fractal like Fibonaccis and powers of 2. But there’s occasional symmetry in the prime fin-digs. I’ve marked some palindromic patterns in red and green:

Palindromic patterns in final 1-digits of the primes in b10 (click for larger)

The palindromic patterns, or pal-pats, in the primes look like the altars of mathness in the triangulars. They’re created by digital palindromes like these:

19, 23, 29 (c=3)
347, 349, 353, 359, 367 (c=5)
937, 941, 947, 953, 967, 971, 977 (c=7)
1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011 (c=9)
26423, 26431, 26437, 26449, 26459, 26479, 26489, 26497, 26501, 26513 (c=10)

Here are the first few pal-pats in the primes (note that 157, 163, 167 and 163, 167, 173 overlap):

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607…

And are there palindromes among the final 2-digits, 3-digits and higher n-digits of the primes in different bases? Yes, you can easily find some. But I haven’t put them on a graph yet:

Base 10 (2-dig)

58789, 58831, 58889 (c=3)
286873, 286927, 286973 (c=3)
360649, 360653, 360749 (c=3)
404851, 404941, 404951 (c=3)
590437, 590489, 590537 (c=3)
623071, 623107, 623171 (c=3)
651517, 651587, 651617 (c=3)

Base 6 (2-dig)

300335, 300401, 300441, 300501, 300535 (c=5) (23459 to 23531 in base 10)
1030255, 1030331, 1030351, 1030431, 1030455 (c=5) (50651 to 50723 in b10)
1140451, 1140501, 1140521, 1141001, 1141051 (c=5) (59791 to 59863 in b10)
1402451, 1402545, 1403031, 1403045, 1403051 (c=5) (78367 to 78439 in b10)
1435431, 1435451, 1435505, 1435551, 1440031 (c=5) (82891 to 82963)
2400505, 2401001, 2401015, 2401101, 2401105 (c=5) (124601 to 124673)
2442235, 2442311, 2442351, 2442411, 2442435 (c=5) (130127 to 130199)
2444215, 2444225, 2444311, 2444325, 2444415 (c=5) (130547 to 130619)
2533105, 2533121, 2533215, 2533221, 2533305 (c=5) (136769 to 136841)

Base 4 (3-dig)

20013013, 20013133, 20020013 (c=3) (33223 to 33287 in base 10)
21031111, 21031303, 21032111 (c=3) (37717 to 37781)
22310011, 22310333, 22311011 (c=3) (44293 to 44357)
33030121, 33031001, 33031121 (c=3) (62233 to 62297)
102031333, 102032131, 102032333 (c=3) (74623 to 74687)
110013121, 110013311, 110020121 (c=3) (82393 to 82457)

Base 3 (3-dig)

112121020012, 112121021211, 112121022021, 112121100211, 112121101012 (c=5) (287393 to 287501 in base 10)
202002212002, 202002212101, 202002220001, 202002221101, 202010000002 (c=5) (395741 to 395849)
1001012111212, 1001012112202, 1001012121022, 1001012121202, 1001012122212 (c=5) (555143 to 555251)
1010112112012, 1010112112201, 1010112120222, 1010112121201, 1010112200012 (c=5) (601079 to 601187)
1011202211211, 1011202212212, 1011202220022, 1011202221212, 1011202222211 (c=5) (625369 to 625477)

Base 2 (5-dig)

101110001111101, 101110010000111, 101110010001001, 101110010100111, 101110010111101 (c=5) (23677 to 23741 in base 10)
10000001000111101, 10000001001011001, 10000001001110001, 10000001001111001, 10000001001111101 (c=5) (66109 to 66173)
10111100110011011, 10111100110011111, 10111100110111001, 10111100110111111, 10111100111011011 (c=5) (96667 to 96731)
11010000111110001, 11010001000001101, 11010001000011001, 11010001000101101, 11010001000110001 (c=5) (106993 to 107057)

And I conjecture that you’ll get palindromes for any number of final digits in all bases. And can these palindromes be of arbitrary length? Again, I conjecture so. There are infinitely many primes and very rare patterns can occur infinitely often in an infinite set of numbers.

Post-Performative Post-Scriptum

Here’s Dan Seagrave’s classic cover for Morbid Angel’s Altars of Madness (1989):

• Morbid Angel — official website
• Dan Seagrave — official website

Elsewhere Other-Accessible…

• Formulas Focal to the Flesh — a pre-previous post paronomasizing the title of a Morbid-Angel album…

Red Sails in the Subset

by in Arithmetic, Geometry, Mathematics, Reciprocals and tagged , , , , , , , , , , , , , | Leave a comment

Let’s look at a simple arithmetical rule and a simple arithmetical fact. And the complexity they can create. First the rule. Subtracting a negative number is the same as adding the positive form of that number:

7 – +2 = 5
7 – -2 = 7 + 2 = 9

-10 – +4 = -14
-10 – -4 = -10 + 4 = -6

Now the simple arithmetical fact: The reciprocal of positive x, namely 1/x, is less than 1 when x > 1, identical to x when x = 1, and greater than 1 when 0 < x < 1. Negative x, -x, works in the opposite direction:

1/5 = 0.2; 1/-5 = -0.2
1/4 = 0.25; 1/-4 = -0.25
1/3 = 0.333333…; 1/-3 = -0.333333…
1/2 = 0.5; 1/-2 = -0.5

1/1 = 1; 1/-1 = -1

1/0.5 = 2; 1/-0.5 = -2
1/0.25 = 4; 1/-0.25 = -4
1/0.333333.. = 3; 1/-0.333333.. = -3
1/0.2 = 5; 1/-0.2 = -5

Now, the simple arithmetical rule and the simple arithmetical fact explain the wildly different behaviour of these two nearly identical formulae:

Formula #1: x = x + 1/x
Formula #2: x = x – 1/x

If you seed x = x + 1/x with 2, this is what happens:

2 = x
2.5 = 2 + 1/2 = 2 + 0.5
2.9 = 2.5 + 1/2.5 = 2.5 + 0.4
3.244827586206896551724137931… = 2.9 + 1/2.9 = 2.9 + 0.3448275862…
3.553010370478947561288431236…
3.834461842815967366750790750…
4.095254632258778985771918456…
4.339439692724345181049239663…
4.569884190357676650018985962…
4.788708116379690742064597208…
4.997532704493448986664559639…
5.197631445038131469095668466…
5.390026771750770995914851381…
5.575554607204394029915651664…
5.754908962142979073283550015…
5.928673657045750549124213874…
6.097345447373015508408978797…
6.261351244425377152997703626…
6.421061179383957004641284553…
6.576798676981813718180627345…

The value of x steadily (but more and more slowly) increases. But when you seed the other formula, x = x – 1/x, with 2, this is what happens:

+2
+1.5 = 2 – 1/2 = 2 – 0.5
+0.8333333… = 1.5 – 1/1.5 = 1.5 – 0.666666…
-0.3666666… = 0.8333333… – 1/0.8333333… = 0.8333333… – 1.2
+2.3606060606… = -0.3666666… – 1/-0.3666666… = -0.3666666… – -2.72727272… = -0.3666666… + 2.72727272…
+1.936986034932119656124790913…
+1.420720051612810742016492942…
+0.716851616121389735975863550…
-0.678137217705362317788764881…
+0.796490591963802485322149292…
-0.459017018658980935029501857…
+1.719551442531198550688634398…
+1.138004432499332885157841729…
+0.259273233005005595158072588…
-3.597661740227243739940039228…
-3.319703423907923593779727545…
-3.018471695555874174383708009…
-2.687178213005645221877765061…
-2.315040631969854351245993463…
-1.883082770759830608578236571…
-1.352038668223383718148747858…
-0.612414851610188982350276645…
+1.020465208974159220420492697…
+0.040519992610273807119693182…
-24.63865528804984441050796942…
-24.59806865747650234381633987…
-24.55741505926418687092326558…
-24.51669416101057552476382150…
-24.47590562755526483917345018…
-24.43504912094763238695385804…

The value of x swings between positive and negative in an irregular, non-periodic way, alternating between slow deterministic decay and instantaneous jumps to sometimes large positive or negative values. The deterministic decays explains why, as we’ll see, there are beautiful regular curves — parabolic curves — amid the irregularity. When the function creates a positive number x > 1, it nibbles away at x until x x > -1, x becomes positive at the next step and the process continues. Represented as a graph, x = x – 1/x looks like this when seeded with 2 — note the parabolic curves:

x[i] = x[i-1] – 1/x[i-1], x[1] = 2 (click for larger)

A shark-fin and some red sails (images StockCake + Para-Sailing World Championship)

Sydney Opera House (image Wikipedia)

When x > 0, its value is represented in white; when x < 0, its value is represented in red. The curves created remind of me of shark-fins or sails or Sydney Opera House. So you could say the graph contains red sails in the subset, i.e. the set of values of x that are sub-zero. Here are some variations on the formula:

x = x – (1/4)/x, x[1] = 2

x = x – (4/3)/x, x[1] = 2

x = x – (4/5)/x, x[1] = 2

Now try this formula, x = 1 – 1/x. When it’s seeded with 2, it behaves like this:

2
0.5 = 1 – 1/2 = 1 – 0.5
-1 = 1 – 1/0.5 = 1 – 2
2 = 1 – -1/-1 = 1 – -1 = 1 + 1
1/2
-1
2
[…]

The values cycles through 2, 0.5, -1, 2, 0.5… for ever. So try varying the formula. This is what happens with x = 0.1 – 1.7/x, seeded with 2:

+2
-0.75
+2.366666666666666666666666666…
-0.618309859154929577464788732…
+2.849430523917995444191343963…
-0.496610440482852346310656327…
+3.523206323143542441364433927…
-0.382515028663775083373274222…
+4.544269826308639632084352464…
-0.274097504104620631734792447…
+6.302172491695235560374503419…
-0.169748249867834571832745868…
+10.11483079397645866692882142…
-0.068070038404634195480677189…
+25.07427708053510247498559239…

When you look at the graph of x = 0.1 – 1.7/x, you’ll see it’s also cycling, just in a more complicated way:

x = 0.1 – 1.7/x, x[1] = 2 (click for larger)

And here’s how different seeds can change the graph:

x = 2/3 – 1/x, x[1] = 2/3

x = 2/3 – 1/x, x[1] = 3/2

This graph reminds me of vertebrae:

x = 2/5 – 1/x, x[1] = 2

And this graph reminds of a bone:

x = 9/7 – 1/x, x[1] = 2

As Lucretius nearly said: Mathematica Moles et Machina Mundi — Mathematics is the Mass and Body of the World.

Elsewhere Other-Accessible…

• Moto-Motto — what Lucretius did say