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

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 [...]

Fingers are fractal. Where a tree has a trunk, branches and twigs, a human being has a torso, arms and fingers. And human beings move in fractal ways. We use our legs to move large distances, then reach out with our arms over smaller distances, then move our fingers over smaller distances still. We’re fractal beings, inside and out, brains and blood-vessels, fingers and toes.

But fingers are fractal are in another way. A digit – digitus in Latin – is literally a finger, because we once counted on our fingers. And digits behave like fractals. If you look at numbers, you’ll see that they contain patterns that echo each other and, in a sense, recur on smaller and smaller scales. The simplest pattern in base 10 is (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). It occurs again and again at almost very point of a number, like a ten-hour clock that starts at zero-hour:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9…
10, 11, 12, 13, 14, 15, 16, 17, 18, 19…
200… 210… 220… 230… 240… 250… 260… 270… 280… 290…

These fractal patterns become visible if you turn numbers into images. Suppose you set up a square with four fixed points on its corners and a fixed point at its centre. Let the five points correspond to the digits (1, 2, 3, 4, 5) of numbers in base 6 (not using 0, to simplify matters):

1, 2, 3, 4, 5, 11, 12, 13, 14, 15, 21, 22, 23, 24, 25, 31, 32, 33, 34, 35, 41, 42, 43, 44, 45, 51, 52, 53, 54, 55, 61, 62, 63, 64, 65… 2431, 2432, 2433, 2434, 2435, 2441, 2442, 2443, 2444, 2445, 2451, 2452…

Move between the five points of the square by stepping through the individual digits of the numbers in the sequence. For example, if the number is 2451, the first set of successive digits is (2, 4), so you move to a point half-way between point 2 and point 4. Next come the successive digits (4, 5), so you move to a point half-way between point 4 and point 5. Then come (5, 1), so you move to a point half-way between point 5 and point 1.

When you’ve exhausted the digits (or frigits) of a number, mark the final point you moved to (changing the colour of the pixel if the point has been occupied before). If you follow this procedure using a five-point square, you will create a fractal something like this:
$fractal4_1single$

$fractal4_1$
A pentagon without a central point using numbers in a zero-less base 7 looks like this:
$fractal5_0single$

$fractal5_0$
A pentagon with a central point looks like this:
$fractal5_1single$

$fractal5_1$
Hexagons using a zero-less base 8 look like this:
$fractal6_1single$

$fractal6_1$

$fractal6_0single$

$fractal6_0$
But the images above are just the beginning. If you use a fixed base while varying the polygon and so on, you can create images like these (here is the program I used):
$fractal4$

$fractal5$

$fractal6789$

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

მეფე (2) • მსოფლიოს (3) • მათემატიკა (5) •

Tag Archives: nonagons

Fine as Nine

Fingering the Frigit