Fine as Nine

Thursday, 20th Jul, 2023 by Krilling for Company in Geometry, Integer Sequences, Mathematics, Polygons, Reciprocals, Triangular Numbers and tagged enneagonal numbers, icosikaihenagonal, Integer Sequences, nonagons, pentadecagonal numbers, Polygonal numbers, reciprocals, recreational math, recreational mathematics, recreational maths, Triangular Numbers | Leave a comment

This is a regular nonagon (a polygon with nine sides):

A nonagon or enneagon (from Wikipedia)

And this is the endlessly repeating decimal of the reciprocal of 7:

1/7 = 0.142857142857142857142857…

What is the curious connection between 1/7 and nonagons? If I’d been asked that a week ago, I’d’ve had no answer. Then I found a curious connection when I was looking at the leading digits of polygonal numbers. A polygonal number is a number that can be represented in the form of a polygon. Triangular numbers look like this:

* = 1
* ** = 3 * ** *** = 6 * ** *** **** = 10
* ** *** **** ***** = 15

By looking at the shapes rather than the numbers, it’s easy to see that you generate the triangular numbers by simply summing the integers:

1 = 1 1+2=3 1+2+3=6 1+2+3+4=10 1+2+3+4+5=15

Now try the square numbers:

* = 1
** ** = 4 *** *** *** = 9 **** **** **** **** = 16 ***** ***** ***** ***** ***** = 25

You generate the square numbers by summing the odd integers:

1 = 1 1+3 = 4 1+3+5 = 9 1+3+7 = 16 1+3+7+9 = 25

Next come the pentagonal numbers, the hexagonal numbers, the heptagonal numbers, and so on. I was looking at the leading digits of these numbers and trying to find patterns. For example, when do the leading digits of the k-th triangular number, tri(k), match the digits of k? This is when:

tri(1) = 1 tri(19) = 190 tri(199) = 19900 tri(1999) = 1999000 tri(19999) = 199990000 tri(199999) = 19999900000 [...]

That pattern is easy to explain. The formula for the k-th polygonal number is k * ((pn-2)*k + (4-pn)) / 2, where pn = 3 for the triangular numbers, 4 for the square numbers, 5 for the pentagonal numbers, and so on. Therefore the k-th triangular number is k * (k + 1) / 2. When k = 19, the formula is 19 * (19 + 1) / 2 = 19 * 20 / 2 = 19 * 10 = 190. And so on. Now try the pol(k) = leaddig(pol(k)) for higher polygonal numbers. The patterns are easy to predict until you get to the nonagonal numbers:

square(10) = 100 square(100) = 10000 square(1000) = 1000000 square(10000) = 100000000 square(100000) = 10000000000 [...]
pentagonal(7) = 70 pentagonal(67) = 6700 pentagonal(667) = 667000 pentagonal(6667) = 66670000 pentagonal(66667) = 6666700000 [...] hexagonal(6) = 66 hexagonal(51) = 5151 hexagonal(501) = 501501 hexagonal(5001) = 50015001 hexagonal(50001) = 5000150001 [...] heptagonal(5) = 55 heptagonal(41) = 4141 heptagonal(401) = 401401 heptagonal(4001) = 40014001 heptagonal(40001) = 4000140001 [...] octagonal(4) = 40 octagonal(34) = 3400 octagonal(334) = 334000 octagonal(3334) = 33340000 octagonal(33334) = 3333400000 [...]
nonagonal(4) = 46 nonagonal(30) = 3075 nonagonal(287) = 287574 nonagonal(2858) = 28581429 nonagonal(28573) = 2857385719 nonagonal(285715) = 285715000000 nonagonal(2857144) = 28571444285716 nonagonal(28571430) = 2857143071428575 nonagonal(285714287) = 285714287571428574 nonagonal(2857142858) = 28571428581428571429 nonagonal(28571428573) = 2857142857385714285719 nonagonal(285714285715) = 285714285715000000000000 nonagonal(2857142857144) = 28571428571444285714285716 nonagonal(28571428571430) = 2857142857143071428571428575 nonagonal(285714285714287) = 285714285714287571428571428574 nonagonal(2857142857142858) = 28571428571428581428571428571429 nonagonal(28571428571428573) = 2857142857142857385714285714285719 nonagonal(285714285714285715) = 285714285714285715000000000000000000 nonagonal(2857142857142857144) = 28571428571428571444285714285714285716 nonagonal(28571428571428571430) = 2857142857142857143071428571428571428575 [...]

What’s going on with the leading digits of the nonagonals? Well, they’re generating a different reciprocal. Or rather, they’re generating the multiple of a different reciprocal:

1/7 * 2 = 2/7 = 0.285714285714285714285714285714...

And why does 1/7 have this curious connection with the nonagonal numbers? Because the nonagonal formula is k * (7k-5) / 2 = k * ((9-2) * k + (4-pn)) / 2. Now look at the pentadecagonal numbers, where pn = 15:

pentadecagonal(1538461538461538461540) = 15384615384615384615406923076923076923076930
2/13 = 0.153846153846153846153846153846...
pentadecagonal formula = k * (13k - 11) / 2 = k * ((15-2)*k + (4-15)) / 2

Penultimately, let’s look at the icosikaihenagonal numbers, where pn = 21:

icosikaihenagonal(2) = 21 icosikaihenagonal(12) = 1266 icosikaihenagonal(107) = 107856 icosikaihenagonal(1054) = 10544743 icosikaihenagonal(10528) = 1052878960 icosikaihenagonal(105265) = 105265947385 icosikaihenagonal(1052633) = 10526335263165 icosikaihenagonal(10526317) = 1052631731578951 icosikaihenagonal(105263159) = 105263159210526318 icosikaihenagonal(1052631580) = 10526315801578947370 icosikaihenagonal(10526315791) = 1052631579163157894746 icosikaihenagonal(105263157896) = 105263157896368421052636 icosikaihenagonal(1052631578949) = 10526315789497368421052643 icosikaihenagonal(10526315789475) = 1052631578947542105263157900 icosikaihenagonal(105263157894738) = 105263157894738263157894736845 icosikaihenagonal(1052631578947370) = 10526315789473706842105263157905 icosikaihenagonal(10526315789473686) = 1052631578947368689473684210526331 icosikaihenagonal(105263157894736843) = 105263157894736843000000000000000000 icosikaihenagonal(1052631578947368422) = 10526315789473684220526315789473684211 icosikaihenagonal(10526315789473684212) = 1052631578947368421257894736842105263166
2/19 = 0.1052631578947368421052631579
icosikaihenagonal formula = k * (19k - 17) / 2 = k * ((21-2)*k + (4-21)) / 2

And ultimately, let’s look at this other pattern in the leading digits of the triangular numbers, which I can’t yet explain at all:

tri(904) = 409060 tri(6191) = 19167336 tri(98984) = 4898965620 tri(996694) = 496699963165 tri(9989894) = 49898996060565 tri(99966994) = 4996699994681515 tri(999898994) = 499898999601055515 tri(9999669994) = 49996699999451815015 tri(99998989994) = 4999898999960055555015 tri(999996699994) = 499996699999945018150015 tri(9999989899994) = 49999898999996005055550015 tri(99999966999994) = 4999996699999994500181500015 tri(999999898999994) = 499999898999999600500555500015 [...]

Hour Power

Sunday, 11th Jun, 2023 by Krilling for Company in Fractals, Geometry, Mathematics and tagged animated gifs, divide-and-discard, fractals, hourglass fractal | Leave a comment

How do you get an hourglass from this shape?

Rep-4 L-tromino

In fact, it’s easy. You simply divide the shape into four identical copies of itself, discard one copy, and repeat the process with each of the sub-copies:

l-triomino_124

Constructing an hourglass (animated)

↓

l-triomino_124_upright_static1

Hourglass (static)

Here are some more posts about what I call the hourglass fractal:

• The Hourglass Fractal at Overlord of the Über-feral

Tright Sights

Monday, 5th Jun, 2023 by Krilling for Company in Circles, Fractals, Geometry, Mathematics, Square Roots, Triangles and tagged animated gifs, √3, circular fractals, fractal geometry, right triangles, square root of 3, triangular fractals | Leave a comment

Here’s a right triangle, where a^2 + b^2 = c^2. But what are the exact values of a, b, and c?

You might be able to guess by eye, but could you prove your guess? Now try the same right triangle tiled with three identical copies of itself:

1-√3-2 triangle as rep3 rep-tile

Now you can prove the exact values of a, b, and c. If the vertical side, a, is 1, then the hypotenuse, c, is 2, because the length that fits once into a fits twice into c. Therefore 2^2 = 1^2 + b^2 → 4 = 1 + b^2 → 4-1 = b^2 → 3 = b^2 → √3 = b. The horizontal side, b, has a length of √3 = 1.73205080757… So the right triangle is 1-√3-2. And if it’s rep3, that is, can be divided into three identical copies of itself, then it’s also rep9, rep27, and so on:

1-√3-2 triangle as rep9 rep-tile

1-√3-2 triangle as rep27 rep-tile

1-√3-2 triangle as rep81 rep-tile

1-√3-2 triangle as rep243 rep-tile

1-√3-2 triangle as rep729 rep-tile

Once you’ve got a rep-tile, you can create fractals. But the 1-√3-2 triangle is cramped. You need more space to work with. And it’s easy to find that space when you realize that a standard equilateral triangle can be divided into six 1-√3-2 triangles:

Equilateral triangle divided into six 1-√3-2 triangles

Equilateral triangle tiled with 1-√3-2 triangles (stage 1)

(please open in new window if image is distorted)

Equilateral triangle tiled with 1-√3-2 triangles (stage 2)

Equilateral triangle tiled with 1-√3-2 triangles (stage 3)

Here are variant colorings of the stage-3 tiled triangle:

But where are the fractals? In one way, you’ve already seen them. But they get more obvious like this:

Fractal stage 1

Fractal #2

Fractal #3

Fractal #4

Fractal #5

Fractal #6

Fractal #7

Fractal #8

Fractal (animated)

Another fractal stage 1

[…]

↓

[…]

Another fractal #8

Another fractal (animated)

And when you have a fractal created using an equilateral triangle, it’s easy to expand the fractal into a circle, like this:

Original fractal

↓

Fractal expanded into circle

Triangular fractal to circular fractal (animated)

Extra Tetra

Monday, 29th May, 2023 by Krilling for Company in Fractals, Geometry, Mathematics and tagged animated gifs, fractals, math, mathematics, maths, polyhedra, polyhedron, polyhedrons, regular polyhedra, Sierpiński tetrahedron, tetrahedra, tetrahedron, tetrahedrons | Leave a comment

Construction of a Sierpiński tetrahedron (from WikiMedia)

Post-Performative Post-Scriptum

The toxic title of this incendiary intervention radically references George Harrison’s album Extra Texture (1975).

First Whirled Warp

Friday, 19th May, 2023 by Krilling for Company in Circles, Geometry, Mathematics, Squares, Triangles and tagged circle, circumference, geometry, midpoints, perimeter, Polygons, rotating points | Leave a comment

Imagine two points moving clockwise around the circumference of a circle. Find the midpoint between the two points when one point is moving twice as fast as the other. The midpoint will trace this shape:

Midpoint of two points moving around circle at speeds s and s*2

(n.b. to make things easier to see, the red circle shown here and elsewhere is slightly larger than the virtual circle used to calculate the midpoints)

Now suppose that one point is moving anticlockwise. The midpoint will now trace this shape:

Midpoint for s, -s*2

Now try three points, two moving at the same speed and one moving twice as fast:

Midpoint for s, s, s*2

When the point moving twice as fast is moving anticlockwise, this shape appears:

Midpoint for s, s, -s*2

Here are more of these midpoint-shapes:

Midpoint for s, s*3

Midpoint for s, -s*3

Midpoint for s*2, s*3

Midpoint for s, -s, s*2

Midpoint for s, s*2, -s*2

Midpoint for s, s*2, s*2

Midpoint for s, -s*3, -s*5

Midpoint for s, s*2, s*3

Midpoint for s, s*2, -s*3

Midpoint for s, -s*3, s*5

Midpoint for s, s*3, s*5

Midpoint for s, s, s, s*3

Midpoint for s, s, s, -s*3

Midpoint for s, s, -s, s*3

Midpoint for s, s, -s, -s*3

But what about points moving around the perimeter of a polygon? Here are the midpoints of two points moving clockwise around the perimeter of a square, with one point moving twice as fast as the other:

Midpoint for square with s, s*2

And when one point moves anticlockwise:

Midpoint for square with s, -s*2

If you adjust the midpoints so that the square fills a circle, they look like this:

↓

Midpoint for square with s, s*2, with square adjusted to fill circle

When the red circle is removed, the midpoint-shape is easier to see:

Midpoint for square with s, s*2, circ-adjusted

Here are more midpoint-shapes from squares:

Midpoint for s, s*3

Midpoint for s, -s*3

Midpoint for s, s*4

And some more circularly adjusted midpoint-shapes from squares:

Midpoint for s, s*3, circ-adjusted

Midpoint for s*2, s*3, circ-adjusted

Midpoint for s, s*5, circ-adjusted

Midpoint for s, s*6, circ-adjusted

Midpoint for s, s*7, circ-adjusted

Finally (for now), let’s look at triangles. If three points are moving clockwise around the perimeter of a triangle, one moving four times as fast as the other two, the midpoint traces this shape:

Midpoint for triangle with s, s, s*4

Now try one of the points moving anticlockwise:

Midpoint for s, s, -s*4

Midpoint for s, -s, s*4

If you adjust the midpoints so that the triangular space fills a circle, they look like this:

Midpoint for s, s, s*4, with triangular space adjusted to fill circle

Midpoint for s, -s, s*4, circ-adjusted

Midpoint for s, s, -s*4, circ-adjusted

There are lots more (infinitely more!) midpoint-shapes to see, so watch this (circularly adjusted) space.

Elsewhere Other-Accessible…

• Second Whirled Warp — more on points moving around polygons
• Third Whirled Warp — even more on points moving around polygons
• We Can Circ It Out — more on converting polygons into circles

Strange “S” in the Light

Wednesday, 23rd Nov, 2022 by Krilling for Company in Fractals, Geometry, Mathematics and tagged fractals, recreational math, recreational mathematics, recreational maths | Leave a comment

Unexpected discoveries are one of the joys of mathematics, even for amateurs. And computers help you make more of them, because computers make it easy to adjust variables or search faster and further through math-space than any unaided human ever could (on the downside, computers can make you lazy and blunt your intuition). Here’s an unexpected discovery I made using a computer in November 2020:

A distorted and dissected capital “S”

It’s a strange “S” that looks complex but is constructed very easily from three simple lines. And it’s a fractal, a shape that contains copies of itself at smaller and smaller scales:

Five sub-fractals of the Strange “S”

Elsewhere Other-Accessible…

• Fractangular Frolics — the Strange “S” in more light

We Can Circ It Out

Saturday, 5th Nov, 2022 by Krilling for Company in Circles, Fractals, Geometry, Mathematics, Polygons, Squares, Triangles and tagged animated gifs, Circles, circular fractals, fractal geometry, fractals, geometry, math, mathematics, maths, Polygons, recreational math, recreational mathematics, recreational maths, Sierpinski triangles, Sierpiński triangle | Leave a comment

It’s a pretty little problem to convert this triangular fractal…

Sierpiński triangle (Wikipedia)

…into its circular equivalent:

Sierpiński triangle as circle

Sierpiński triangle to circle (animated)

But once you’ve circ’d it out, as it were, you can easily adapt the technique to fractals based on other polygons:

T-square fractal (Wikipedia)
⇓

T-square fractal as circle

T-square fractal to circle (animated)

Elsewhere other-accessible…

• Dilating the Delta — more on converting polygonic fractals to circles…

Diamonds to Dust

Tuesday, 13th Sep, 2022 by Krilling for Company in Circles, Fractals, Geometry, Hexagons, Mathematics, Triangles and tagged animated gifs, diamonds, divide-and-discard, equilateral triangles, fractal geometry, fractals, geometry, Hexagons, math, mathematics, maths, Mitsubishi diamonds, Mitsubishi logo, recreational math, recreational mathematics, recreational maths | Leave a comment

It’s one of the most famous and easily recognizable logos in the world:

The Mitsubishi diamonds (source)

Those are the three diamonds of Mitsubishi, whose name itself means “three diamonds” or “three rhombi” in Japanese (see 三菱). But if you look at the Mitsubishi diamonds with a mathematical eye, you can see how to create them in two simple steps. First, you divide an equilateral triangle into nine smaller equilateral triangles. Then you discard three of the sub-triangles, like this:

Equilateral triangle divided into nine sub-triangles

↓

Six sub-triangles left after three are discarded

↓

But why stop there? Once you’ve discarded three triangles, six triangles are left. Now do the same to the remaining six: divide each into nine sub-triangles and discard three of the sub-triangles. Then do it again and again. When you’ve reduced the diamonds to dust, you’ve got a fractal, a shape that repeats itself at smaller and smaller scales:

Diamond fractal stage #1

Diamond fractal stage #2

Diamond fractal stage #3

Diamond fractal stage #4

Diamond fractal stage #5

Diamond fractal stage #6

Diamond fractal (animated)

After that, you can convert the fractal-within-a-triangle into a fractal-within-a-circle:

↓

Diamond fractal, triangular to circular (animated)

You can create other fractals by dividing-and-discarding sub-triangles from a rep-9 equilaterial triangle. Here’s what I call a rep9-tri grid fractal:

Grid fractal stage #1

Grid fractal stage #2

Grid fractal stage #3

Grid fractal stage #4

Grid fractal stage #5

Grid fractal stage #6

Grid fractal stage #7

Grid fractal (animated)

↓

Grid fractal, triangular to circular (animated)

And here’s a rep9-tri hexagon fractal:

Hexagon fractal (initial form)

Hexagon fractal stage #1

Hexagon fractal stage #2

Hexagon fractal stage #3

Hexagon fractal stage #4

Hexagon fractal stage #5

Hexagon fractal stage #6

Hexagon fractal (animated)

↓

Hexagon fractal, triangular to circular (animated)

The Belles of El

Sunday, 11th Sep, 2022 by Krilling for Company in Geometry, History, Language, Mathematics and tagged ancient Greek mathematics, Arithmetica, Astronomia, beautiful women, beautiful young women, early books, early modern English, English language, Euclid, Euclid's Elements, Geometria, geometry, Greek geometry, math, mathematics, maths, Musica, personification, Sir Henry Billingsley, The Elements of Geometrie of the Most Aucient Philosopher Evclide of Megara, young women | Leave a comment

Title page of Sir Henry Billingsley’s first English version of Euclid’s Elements, 1570, with personifications of Geometria, Astronomia, Arithmetica and Musica as beautiful young women

The Elements of Geometrie of the Moſt Aucient Philoſopher Evclide of Megara.

Faithfully (now first) tranʃlated into the Engliʃhe toung, by H. Billingſley, Citizen of London.

Whereunto are annexed certaine Scolies, Annotations, and Inuentions, of the best Mathematiciens, both of times past, and in this our age.

With a very fruitfull Præface made by M.I. Dee, ʃpecifying the chiefe Mathematicall Sciences, what they are, and wherunto commodious: where, alʃo, are diʃcloʃed certaine new Secrets Mathematicall and Mechanicall, untill theʃe our daies, greatly miʃʃed.

Imprinted at London by Iohn Daye.

The title of this incendiary intervention is a paronomasia on “The Bells of Hell…”, a British airmen’s song in terms of core issues around World War I.

Fylfy Fractals

Friday, 19th Aug, 2022 by Krilling for Company in Fractals, Geometry, Mathematics, Rep-Tiles, Squares, Triangles and tagged animated gifs, cross cramponnée, fractals, fylfot, gammadion, Hakenkreuz, math, mathematics, maths, recreational mathematics, swastika | Leave a comment

An equilateral triangle is a rep-tile, because it can be tiled completely with smaller copies of itself. Here it is as a rep-4 rep-tile, tiled with four smaller copies of itself:

Equilateral triangle as rep-4 rep-tile

If you divide and discard one of the sub-copies, then carry on dividing-and-discarding with the sub-copies and sub-sub-copies and sub-sub-sub-copies, you get the fractal seen below. Alas, it’s not a very attractive or interesting fractal:

Divide-and-discard fractal stage #1

Stage #2

Stage #3

Stage #4

Stage #5

Stage #6

Stage #7

Stage #8

Stage #9

Divide-and-discard fractal (animated)

You can create more attractive and interesting fractals by rotating the sub-triangles clockwise or anticlockwise. Here are some examples:

Now try dividing a square into four right triangles, then turning each of the four triangles into a divide-and-discard fractal. The resulting four-fractal shape is variously called a swastika, a gammadion, a cross cramponnée, a Hakenkreuz and a fylfot. I’m calling it a fylfy fractal:

Divide-and-discard fractals in the four triangles of a divided square stage #1

Fylfy fractal #2

Fylfy fractal #3

Fylfy fractal #4

Fylfy fractal #5

Fylfy fractal #6

Fylfy fractal #7

Fylfy fractal #8

Fylfy fractal (animated)

Finally, you can adjust the fylfy fractals so that each point in the square becomes the equivalent point in a circle:

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

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Category Archives: Geometry