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What is infinity actually?
What is infinity actually?
The science behind infinity! Fascinating isn’t it? You may have heard the word infinity, several times in your life. And maybe you understand what it is. Or do you? Humans have always known about it from distant history. There are records that early Greek Philosophers like Anaximander (610 – 546 BC), used the word apeiron, to state something which is unbounded and indefinite. The great Aristotle…
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Some dumb math stuff
I found a parametrization for all Pythagorean triples such that c-b=1:
(2n+1)^2 + (2n^2+2n)^2 = (2n^2+2n+1)^2
This makes familiar triples like the 3-4-5 and the 5-12-13. I also made a parametrization for triples such that c-b=2.
At this point I'm just hoping there's a pattern I can extract for a general difference of k between c and b. Then I'd have countably many formulas which necessarily account for EVERY possible Pythagorean triple. 🤞
(Mad scientist mode: engaged) Then I could combine all of these formulas and parametrize both variables using only natural numbers, thus mapping the naturals to ALL PYTHAGOREAN TRIPLES! *Maniacal laughter*
Source.
"When it was proclaimed that the Library contained all books, the first impression was one of extravagant happiness," wrote Jorge Luis Borges in his classic of philosophical fiction, "The Library o...
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Each snowflake is algorithmically generated using some randomness to create infinitely many snowflakes where no two are exactly alike.
Mathematica code:
rr[n_] := (SeedRandom[n]; RandomReal[]) H = Table[{Cos[n*Pi/3], Sin[n*Pi/3]}, {n, 0, 5, 1}]; SnowFlake[Q_, x_, y_, R_, S_, k_, h_, o_, s_, N_, PR_, IS_] := Graphics[{ Rotate[ Translate[ Scale[ Table[ Table[ Rotate[ Translate[ Scale[ Table[ {AbsoluteThickness[k*h^(n - 1)], Opacity[o], White, Line[ {{0, 0}, H[[i]]}]}, {i, 1, 6, 1}], s^(n - 1)], {If[n == 1, 0, rr[Q*n]], 0}], If[n == 1, 0, (j + rr[Q*n])*Pi/3], {0, 0}], {j, 0, 5, 1}], {n, 1, N, 1}], S], {x, y}], R, {x, y}]}, PlotRange -> PR, ImageSize -> IS, Background -> Black] Manipulate[ SnowFlake[Q, 0, 0, rr[2 Q] Pi/3, 1, k, h, o, s, N, 2, 500], {Q, 1, 1000, 1}, {{k, 1}, 0, 2}, {{h, .9}, 1, 0}, {{o, .75}, 1, 0}, {{s, .75}, 1, 0}, {{N, 10}, 1, 20, 1}] Manipulate[ GraphicsGrid[ Table[ SnowFlake[Q*W, 0, 0, (-1)^(Round[rr[4 Q*W]]) (t + rr[2 Q*W]) Pi/3, 1, 1, .85, .8, .5 + .2 rr[3 Q*W], 15, 2, 100], {Q, q, q+6, 1}, {W, w, w+4, 1}], Background -> Black, ImageSize -> {500, 700}, AspectRatio->7/5], {q, 1, 100, 1}, {w, 1, 100, 1}, {t, 0, 1 - 1/25, 1/25}]
Apparently there are two different types of infinity. And here I am getting excited about something as simple as fixed gear bicycles.
Creation of a Sierpinski triangle.
Note: it never ends.
