Performativizing Polyhedra

Τα Στοιχεία του Ευκλείδου, ια΄

κεʹ. Κύβος ἐστὶ σχῆμα στερεὸν ὑπὸ ἓξ τετραγώνων ἴσων περιεχόμενον.
κϛʹ. ᾿Οκτάεδρόν ἐστὶ σχῆμα στερεὸν ὑπὸ ὀκτὼ τριγώνων ἴσων καὶ ἰσοπλεύρων περιεχόμενον.
κζʹ. Εἰκοσάεδρόν ἐστι σχῆμα στερεὸν ὑπὸ εἴκοσι τριγώνων ἴσων καὶ ἰσοπλεύρων περιεχόμενον.
κηʹ. Δωδεκάεδρόν ἐστι σχῆμα στερεὸν ὑπὸ δώδεκα πενταγώνων ἴσων καὶ ἰσοπλεύρων καὶ ἰσογωνίων περιεχόμενον.

Euclid’s Elements, Book 11

25. A cube is a solid figure contained by six equal squares.
26. An octahedron is a solid figure contained by eight equal and equilateral triangles.
27. An icosahedron is a solid figure contained by twenty equal and equilateral triangles.
28. A dodecahedron is a solid figure contained by twelve equal, equilateral, and equiangular pentagons.

Sprime Time

All fans of recreational math love palindromic numbers. It’s mandatory, man. 101, 727, 532235, 8810188, 1367755971795577631 — I love ’em! But where can you go after palindromes? Well, you can go to palindromes in a higher dimension. Numbers like 101, 727, 532235 and 8810188 are 1-d palindromes. That is, they’re palindromic in one dimension: backwards and forwards. But numbers like 181818189 and 646464640 aren’t palindromic in one dimension. They’re palindromic in two dimensions:


1 8 1
8 9 8
1 8 1

n=181818189


6 4 6
4 0 4
6 4 6

n=646464640



They’re 2-d palindromes or spiral numbers, that is, numbers that are symmetrical when written as a spiral. You start with the first digit on the top left, then spiral inwards to the center, like this for a 9-digit spiral (9 = 3×3):


And this for a 36-digit spiral (36 = 6×6):


Spiral numbers are easy to construct, because you can reflect and rotate the numbers in one triangular slice of the spiral to find all the others:


You could say that the seed for the spiral number above is 7591310652, because you can write that number in descending lines, left-to-right, as a triangle.

Here are some palindromic numbers with nine digits in base 3 — as you can see, some are both palindromic numbers and spiral numbers. That is, some are palindromic in both one and two dimensions:

1  0  1

0  1  0

1  0  1

n=101010101


1  0  1

0  2  0

1  0  1

n=101010102


1  1  1

1  0  1

1  1  1

n=111111110


1  1  1

1  1  1

1  1  1

n=111111111


2  0  2

0  1  0

2  0  2

n=202020201


2  0  2

0  2  0

2  0  2

n=202020202


2  2  2

2  1  2

2  2  2

n=222222221


2  2  2

2  2  2

2  2  2

n=222222222


But palindromic primes are even better than ordinary palindromes. Here are a few 1-d palindromic primes in base 10:

101
151
73037
7935397
97356765379
1091544334334451901
1367755971795577631
70707270707
39859395893
9212129
7436347
166000661
313
929


And after 1-d palindromic primes, you can go to 2-d palindromic primes. That is, to spiral primes or sprimes — primes that are symmetrical when written as a spiral:

3 6 3
6 7 6
3 6 3

n=363636367 (prime)
seed=367 (see definition above)


9 1 9
1 3 1
9 1 9

n=919191913 (prime)
seed=913


3 7 8 6 3 6 8 7 3
7 9 1 8 9 8 1 9 7
8 1 9 0 9 0 9 1 8
6 8 0 5 5 5 0 8 6
3 9 9 5 7 5 9 9 3
6 8 0 5 5 5 0 8 6
8 1 9 0 9 0 9 1 8
7 9 1 8 9 8 1 9 7
3 7 8 6 3 6 8 7 3

n=378636873786368737863687378636879189819189819189819189819090909090909090555555557 (prime)
seed=378639189909557 (l=15)


And why stop with spiral numbers — and sprimes — in two dimensions? 363636367 is a 2-sprime, being palindromic in two dimensions. But the digits of a number could be written to form a symmetrical cube in three, four, five and more dimensions. So I assume that there are 3-sprimes, 4-sprimes, 5-sprimes and more out there. Watch this space.

The Power of Powder

• Racine carrée de 2, c’est 1,414 et des poussières… Et quelles poussières ! Des grains de sable qui empêchent d’écrire racine de 2 comme une fraction. Autrement dit, cette racine n’est pas dans Q. — Rationnel mon Q: 65 exercices de styles, Ludmilla Duchêne et Agnès Leblanc (2010)

• The square root of 2 is 1·414 and dust… And what dust! Grains of sand that stop you writing the root of 2 as a fraction. Put another way, this root isn’t in Q [the set of rational numbers].

Thrice Dice Twice

A once very difficult but now very simple problem in probability from Ian Stewart’s Do Dice Play God? (2019):

For three dice [Girolamo] Cardano solved a long-standing conundrum [in the sixteenth century]. Gamblers had long known from experience that when throwing three dice, a total of 10 is more likely than 9. This puzzled them, however, because there are six ways to get a total of 10:

1+4+5; 1+3+6; 2+4+4; 2+2+6; 2+3+5; 3+3+4

But also six ways to get a total of 9:

1+2+6; 1+3+5; 1+4+4; 2+2+5; 2+3+4; 3+3+3

So why does 10 occur more often?

To see the answer, imagine throwing three dice of different colors: red, blue and yellow. How many ways can you get 9 and how many ways can you get 10?

Roll Total=9 Dice #1 (Red) Dice #2 (Blue) Dice #3 (Yellow)
01 9 = 1 2 6
02 9 = 1 3 5
03 9 = 1 4 4
04 9 = 1 5 3
05 9 = 1 6 2
06 9 = 2 1 6
07 9 = 2 2 5
08 9 = 2 3 4
09 9 = 2 4 3
10 9 = 2 5 2
11 9 = 2 6 1
12 9 = 3 1 5
13 9 = 3 2 4
14 9 = 3 3 3
15 9 = 3 4 2
16 9 = 3 5 1
17 9 = 4 1 4
18 9 = 4 2 3
19 9 = 4 3 2
20 9 = 4 4 1
21 9 = 5 1 3
22 9 = 5 2 2
23 9 = 5 3 1
24 9 = 6 1 2
25 9 = 6 2 1
Roll Total=10 Dice #1 (Red) Dice #2 (Blue) Dice #3 (Yellow)
01 10 = 1 3 6
02 10 = 1 4 5
03 10 = 1 5 4
04 10 = 1 6 3
05 10 = 2 2 6
06 10 = 2 3 5
07 10 = 2 4 4
08 10 = 2 5 3
09 10 = 2 6 2
10 10 = 3 1 6
11 10 = 3 2 5
12 10 = 3 3 4
13 10 = 3 4 3
14 10 = 3 5 2
15 10 = 3 6 1
16 10 = 4 1 5
17 10 = 4 2 4
18 10 = 4 3 3
19 10 = 4 4 2
20 10 = 4 5 1
21 10 = 5 1 4
22 10 = 5 2 3
23 10 = 5 3 2
24 10 = 5 4 1
25 10 = 6 1 3
26 10 = 6 2 2
27 10 = 6 3 1

Toxic Turntable #22

Currently listening…

• Dźmutia Zirih, Plz Yrslf (1976)
• Far Beyond Xanadu, Dionysus’ Holy Name (1992)
• Yolanda Grovedrew, Not for Duke War (1997)
• Egzotiq, Vous N’Êtes Que (1984)
• Doctor Yacht, Invoke the Geigar (2009)
• Forschung-239, Jisirlo (1995)
• Gary Jophe, Silver Sands (1992)
• მზის მგელი, მგლისთვალება (2008)
• Helios Epoch, Nahtloser Neuntöter (2009)
• WihlhiW, Gaze Fix (1996)
• Ossafracht, Lokomotiv Zinken (2002)
• Vora xMqa, Future Is An Asylum (2015)
• հաց և գինի, Պետրիկոր (2020)
• Floris Nox, God is Caffeinated (1988)
• Phonophoro L.G., El Coro del Abismo (1988)
• Oscar’s Vital Glove, We Hate Tweeve (2003)
• Ecofoxes, When the Hen (1994)
• ბვემწა, ფვიტი ჰმრე (2017)
• Aoatt Leit, Trey Drake (1993)
• Audiosun, Lucus (Non Lucendo) (1995)
• Hildegard von Bingen, Hortus Deliciarum (2018)
• Ikexon, H.M.T. (2014)


Previously pre-posted

Toxic Turntable #1#2#3#4#5#6#7#8#9#10#11#12#13#14#15#16#17#18#19#20#21