Category Archives: Geometry

Angular Trerkel

by in 2SLGBTQ+ Community Issues, Circles, Fractals, Geometry, Mathematics, Sierpiński triangle, Triangles and tagged , , , , | Leave a comment

Here’s the famous Sierpiński triangle, a fractal named after the Polish mathematician Wacław Sierpiński (1882-1969):

A Sierpiński triangle

You can create it by following all the possible paths of a point jumping half-way towards the vertices of the triangle. But what if the point can jump towards the center of the triangle too? Then you get another fractal, one that looks like this:

Sierpiński triangle when point can jump towards center of triangle too

Now, as a keyly committed core component of the trans-entitial community, I wondered whether triangles and triangular fractals might identify as other geometrical shapes…

I mean, what’s to stop a triangle identifying as a circle? Nothing. And obvs, if one of the Sierpiński triangles above identifies as a circle, it is a circle — trans-circles are circles (get over it). That is, there’s no need for Sierpiński surgery.

But what if a trans-circular Sierpiński triangle wanted to affirm its circularity with surgery? What would it look like then? Well, you can use elementary trig based on the angles of points within the triangle to stretch it into what might be called a trircle, i.e. a trans-circular triangle that has affirmed its inner circular identity:

A pre-op trans-circular Sierpiński triangle

🡇

A post-op trans-circular Sierpiński triangle or Sierpiński trircle

Sierpiński triangle → Sierpiński trircle (animated at EZgif)

On the other hand, the trircle might be a pre-op trans-triangular circle that has NOT affirmed its inner triangular identity with surgery. It’s complicated. But if we suppose it’s a post-op trans-circle rather than a pre-op trans-triangle, we could nickname it an Angular Trerkel, punning on the name of the great German Bundeskanzlerin Angela Merkel. Here’s the same Sierpiński surgery on the centered Sierpiński triangle (the second fractal looked at above):

A pre-op trans-circular centered Sierpiński triangle

🡇

A post-op trans-circular centered Sierpiński triangle or centered Sierpiński trircle

Centered Sierpiński triangle → centered Sierpiński trircle (animated at EZgif)

Again, the post-op trans-circular centered Sierpiński triangle might in fact be a pre-op trans-triangular centered Sierpiński circle. When it comes to the trans-entitial community, always remember: In dubio, interrogāte entitatem! — “In case of doubt, ask the entity!”

A Little Hextra

by in Geometry, Hexagons, Integer Sequences, Mathematics and tagged , , , , , , | Leave a comment

• 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386, 562, 794, 1093, 1471, 1941, 2517, 3214, 4048, 5036, 6196, 7547, 9109, 10903, 12951, 15276, 17902, 20854, 24158, 27841, 31931, 36457, 41449, 46938, 52956, 59536, 66712, 74519, 82993, 92171, 102091, 112792, 124314, 136698 — A000127, “Maximal number of regions obtained by joining n points around a circle by straight lines”, at The Online Encyclopedia of Integer Sequences

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)

Size Scatters

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

While I play New Order, let’s view order. Below are some rational fractions ordered by increasing size of denominator (the lower part of the fraction, e.g. the 7 in 3/7) and numerator (the upper part of the fraction, e.g. the 5 in 5/9). Note the positions of 1/2 and 1/7:

1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 1/8, 3/8, 5/8, 7/8, 1/9, 2/9, 4/9, 5/9, 7/9, 8/9, 1/10, 3/10, 7/10, 9/10

Now let’s plot fractions ordered like that on a fract-L, a graph whose axes represent numbers < 1. If both x and y = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5…, you get a line at 45°:

fract-L for x = 1/2, 1/3, 2/3…, y = 1/2, 1/3, 2/3…

If the fractions went to infinity, you’d get a solid line. As it is, you get some interesting splits in the 45° line. And when you plot the fractions like that, it’s easy to see that each point, (x,y), has a unique x and a unique y. But there are other ways to order the rational fractions. Try the same set ordered by the size of the decimal, not the denominator and numerator. In the previous set, 1/2 came before 1/7. Now 1/2 comes after 1/7, because 0.5 = 1/2 > 0.142857… = 1/7:

1/10, 1/9, 1/8, 1/7, 1/6, 1/5, 2/9, 1/4, 2/7, 3/10, 1/3, 3/8, 2/5, 3/7, 4/9, 1/2, 5/9, 4/7, 3/5, 5/8, 2/3, 7/10, 5/7, 3/4, 7/9, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10

Now let’s plot x as the first set of fractions, ordered by size of denominator-and-numerator, and y as the second set of fractions, ordered by the size of the decimal. You’ll see that this kind of size scatters:

fract-L for x = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5…, y = 1/10, 1/9, 1/8, 1/7, 1/6, 1/5, 2/9…

Now the points aren’t compressed into a 1-d line, but beginning to spread in 2-d space. The more fractions you use, the more the points spread. They remind me of papillae on a fractal tongue:

x = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5… < 1/26, y = sizesort(1/2..1/26)

Papillae on a human tongue (image courtesy chatGPT)

x = denumsort(1/2..1/51), y = sizesort(1/2..1/51)

x = denumsort(1/2..101/), y = sizesort(1/2..1/101)

And the points seem to be occurring at the same x or y value. But that’s an artefact of a screen with limited pixels. On an impossible screen with infinite pixels, each (x,y) still has a unique x and unique y. Here are more fract-Ls with more fractions:

x = denumsort(1/2..1/151), y = sizesort(1/2..1/151)

x = denumsort(1/2..1/201), y = sizesort(1/2..1/201)

x = denumsort(1/2..1/251), y = sizesort(1/2..1/251)

finer detail for x = denumsort(1/2..1/251), y = sizesort(1/2..1/251)

x = denumsort(1/2..37/406), y = sizesort(1/2..37/406)

x = denumsort(1/2..1/501), y = sizesort(1/2..1/501)

Finally, to the closing bars of New Order, let’s view order in an animated gif:

x = denumsort(a/b), y = sizesort(a/b) (animated at EZgif)

Post-Performative Post-Scriptum…

In fact, I wasn’t listening to New Order to view order. I’m not a fan of New Order, just a fan of assonance.

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)

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

Strartifacts

by in Arithmetic, Base Behavior, Geometry, Integer Sequences, Mathematics and tagged , , , , , , , , , , , | Leave a comment

Here’s a sequence of decreasing numbers. Which number comes next?

612 → 600 → 594 → 414 → 398 → 182 → ?

It’s 166, because the numbers decrease by the product of their digits higher than 0:

612 – 6*2 = 612 – 12 = 600 → 600 – 6 = 594 → 594 – 5*9*4 = 594 – 180 = 414 → 414 – 4*4 = 414 – 16 = 398 → 398 – 3*9*8 = 398 – 216 = 166

Eventually the sequence will reached 0 and stop. If you want to see how this function looks on a graph, here it is:

x = n <= 3722, f(i) -= digmul(f(i)) → 0 (click for larger)

The graph represents n on the x-axis, with the red circles marking n = 100 and n = 1000. The sequence of falling digit-products is on the y-axis, but the graph has a special feature there. The y-axis is compressed according to the size of n, so that n = 1000 falls to 0 with n -= digmul(n) in the same height as n = 100. Here’s a graph for the same function in base 7:

x = n <= 3722 in base 7, f(i) -= digmul(f(i)) → 0

Now the red circles represent 7^2 = 49, 7^3 = 343, 7^4 = 2401, i.e. 100b7, 1000b7, 10000b7. And you can try other functions for n = n – func(n) = n -= func(n). Here’s a graph for n -= hailstep(n), where hailstep(n) returns the number of steps in the Collatz sequence for n:

x = n <= 3722 in base 7, f(i) -= hailstep(f(i)) → f(i) < 2

You form a Collatz sequence by starting with a whole number and finding the next number according to two rules:

1. If n(i) is divisible by 2, n(i+1) = n(i) / 2
2. If n(i) is not divisible by 2, n(i+1) = n(i) * 3 + 1

So the Collatz sequence for n = 10 looks like this:

10 → 10 / 2 = 5 → 5 * 3 + 1 = 16 → 16 / 2 = 8 → 8 / 2 = 4 → 4 / 2 = 2 → 2 / 2 = 1.

When you reach 1, you stop. So that’s six steps for n = 10. But does every n reach 1 in the end? It’s a very simple question about a very simple function. But nobody knows and nobody can prove that either all numbers do or at least one number doesn’t. The German mathematician Lothar Collatz (1910-90) conjectured that all numbers do reach 1. But it can take a surprisingly long time, even with small n. This is the Collatz sequence for n = 27:

27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

Now some more functions for the y-compressed fall-bands, as I call them. If you use the sum of the factors * powers, you get this:

x = n <= 7422, f(i) -= factpowsum(f(i)) → f(i) < 2

The factpowsum(n) is the sum of the factors multiplied by their powers. For example, 37692 = 2^2 * 3^3 * 349, so factpowsum(4188) = 2*2 + 3*3 + 349*1 = 362. Here’s factpowsum for more n:

x = n <= 14822, f(i) -= factpowsum(f(i)) → f(i) < 2

You can also use the very simple function f(i) -= 1, that is, compress the numbers from n to 1 into the y-gap. But if you do that, you’ll get a completely filled screen:

x = n <= 3722, f(i) -= 1 → f(i) = 0

So you can adjust the color of a pixel according to how many times it’s written to:

x = n <= 1862, f(i) -= 1 → f(i) = 0 (color-adjust)

The patterns in the colors are artifacts of the limited resolution of the screen, so I call these patterns strartifacts = strata + artifacts. Here’s another example:

x = n <= 3722, f(i) -= 1 → f(i) = 0 (color-adjust)

Or adjust the greytone of the pixel:

x = n <= 3722, f(i) -= 1 → f(i) = 0 (greytone-adjust)

And so on (in all cases, you can click for a larger image):

x = n <= 1862, f(i) -= blockmul(f(i)) (multiply run-lengths of same digits) → f(i) < 2

x is triangular(n) = 3 to 1734453, y is 1 < triangular numbers <= n

x = n <= , f(i) -= 1 → f(i) < 2

x = n <= 7442, f(i) -= 1 → f(i) < 2

x = n <= 1862, f(i) -= leaddig(f(i)) → f(i) < 2 # 1

x = n <= , f(i) -= leaddig(f(i)) → f(i) < 2 # 2

x = n <= , f(i) -= trailingdigit(f(i)) + 1 → f(i) < 2

x = n <= , f(i) -= trailingdigit(f(i) in base 5) + 3 → f(i) < 2

for triangular(n) = 3 to 6928503, f(i) -= primes → f(i) < 2


x = n <= 1862, f(i) -= blockmul(f(i) in base 5) (multiply run-lengths of same digits) → f(i) < 2

x = n <= 1862, f(i) -= blockmul(f(i) in base 2) → f(i) < 2

x = n <= 1862, f(i) -= digsum(f(i)) → f(i) < 2

x = n <= 3722, f(i) -= func(x = 1/4 → x < 0, x(1) = 4) → f(i) < 2

x = n <= 932, f(i) -= func(x -= 3/x → x < 0, x(1) = 6) → f(i) < 2

Formulas Focal to the Flesh

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

Here’s an interesting formula:

fr(1) = 1/2; mx = 3
fr(i) = fr(i-1) + 1/fr(i-1)
if fr(i) > mx, fr(i) = fr(i) – mx

Early terms look like this:

0.5, 2.5, 2.9, 3.244827586…, 4.329334628…, 2.081590666…, 2.561992513…, 2.952313716…, 3.291031107…, 3.727089920…, 2.102435627…, 2.578074447…, 2.965960841…, 3.303119709…, 3.602146368…, 2.262872154…, 2.704788415…, 3.074503101…, 13.49676325…, 10.59203071…, 7.723747777…, 4.935444092…, 2.452121378…, 2.859931533…, 3.209590254…, 4.980804482…, 2.485649867…, 2.887959143…, 3.234224430…, 4.503633905…, 2.168689406…

Can you see any patterns emerging? I’d guess not. And I’d guess a thousand more terms wouldn’t help you see any better. It’s hard for humans to see patterns in a jumble of numbers. Our eyes don’t work as well on numbers as on shapes. That’s why you can make that formula focal to the flesh, as it were, by plotting the numbers on a graph. Or part of the numbers, anyway. Suppose you take the fractional parts of each pair of terms and use them to map (x,y) on a FractL (my name for a graph whose arms run from 0 to 1). For example, the terms 4.935444092… and 2.452121378… would yield x = 0.935444092… and y = 0.452121378… (or vice versa). The resultant graph makes the formula focal to the flesh. And it’s replete with patterns:

fr(i)+=1/fr(i-1); if fr(i)>3, fr(i)-=3; x = frac(fr(i)), y = frac(fr(i+1))

I can’t explain the patterns and they may arise from limited precision in the decimal digits. But I like them however they arise. The graph doesn’t change when mx = 4 (although it creates the lines in a different order):

if fr(i)>4, fr(i)-=4

But it does change when mx = 4/3. The lines almost vanish, except for a tiny comet-like mark towards the upper right-hand corner:

if fr(i)>4/3, fr(i)-=4/3

When mx = 7/2, the graph of mx = 3|4 is back in a slightly different form:

if fr(i)>7/2, fr(i)-=7/2

And again with 7/3:

if fr(i)>7/3, fr(i)-=7/3

There’s a big change with 7/4, Most of the lines disappear:

if fr(i)>7/4, fr(i)-=7/4

And only the main lines appear with 9/5:

if fr(i)>9/5, fr(i)-=9/5

And so on till you try fr -= 2/f, as noted below:

if fr(i)>11/5, fr(i)-=11/5

if fr(i)>11/6, fr(i)-=11/6

if fr(i)>15/8, fr(i)-=15/8

if fr(i)>29/15, fr(i)-=29/15

Now try fr += 2/fr and fr += 3/fr. This is what happens:

fr += 2/fr; if fr(i)>3, fr(i)-=3

fr += 2/fr; if fr(i)>8/3, fr(i)-=8/3

fr += 2/fr; if fr(i)>11/4, fr(i)-=11/4

fr += 3/fr; if fr(i)>6, fr(i)-=6

And what about these graphs?

They’re created by seeding a sum, s, with a fraction, then adding more fractions < 1 whose numerators = 1,2,3… and whose denominators are the prime numbers 1, s -= 1. When s > 1, s -= 1. Then you take the fractional parts of s(i) and s(i+1) and graph (x,y) as above.

Post-Performative Post-Scriptum

The title of this post refers to Morbid Angel’s Formulas Fatal to the Flesh (1998). I’ve never heard it, but I like Morbid Angel’s alphabetically alliterative album-titles.

The Bird Dimension

by in Art, Geometry, Mathematics, Optical Illusions, Post-Weird, Surrealism and tagged , , , , , , , | 2 Comments

M.C. Escher, Another World / Andere Wereld (1947)

This is almost my favorite image by Escher. But I’d like a frEscher perspective in it: I think the bird should be looking in the other direction, out into the impossibly overlapping universes, not into the cupola and at the viewer.