Monthly Archives: May 2026

Partition Fission

by in Arithmetic, Mathematics, Partitions and tagged , , , , | Leave a comment

Simple but seductive. That’s how I’d describe partitions. Except that they’re not so simple. There are hidden depths in the task of finding how many ways an integer can be expressed as the sum of smaller integers (and as the sum of itself). Here are the partitions of 4, for example:

4 = 1+3 = 2+2 = 1+1+2 = 1+1+1+1
4 (1) = 1+3 (2) = 2+2 (3) = 1+1+2 (4) = 1+1+1+1 (5), ∴ partcount(4) = 5

There are five partitions of 4, because an integer is counted as its own partition. Accordingly, partcount(4) = 5. But partcount(n) doesn’t return values in a predictable way:

partcount(1) = 1 ← 1
partcount(2) = 2 ← 2 = 1+1
partcount(3) = 3 ← 3 = 1+2 = 1+1+1
partcount(4) = 5 ← 4 = 1+3 = 2+2 = 1+1+2 = 1+1+1+1
partcount(5) = 7 ← 5 = 1+4 = 2+3 = 1+1+3 = 1+2+2 = 1+1+1+2 = 1+1+1+1+1
partcount(6) = 11 ← 6 = 1+5 = 2+4 = 3+3 = 1+1+4 = 1+2+3 = 2+2+2 = 1+1+1+3 = 1+1+2+2 = 1+1+1+1+2 = 1+1+1+1+1+1
partcount(7) = 15 ← 7 = 1+6 = 2+5 = 3+4 = 1+1+5 = 1+2+4 = 1+3+3 = 2+2+3 = 1+1+1+4 = 1+1+2+3 = 1+2+2+2 = 1+1+1+1+3 = 1+1+1+2+2 = 1+1+1+1+1+2 = 1+1+1+1+1+1+1
partcount(8) = 22 ← 8 = 1+7 = 2+6 = 3+5 = 4+4 = 1+1+6 = 1+2+5 = 1+3+4 = 2+2+4 = 2+3+3 = 1+1+1+5 = 1+1+2+4 = 1+1+3+3 = 1+2+2+3 = 2+2+2+2 = 1+1+1+1+4 = 1+1+1+2+3 = 1+1+2+2+2 = 1+1+1+1+1+3 = 1+1+1+1+2+2 = 1+1+1+1+1+1+2 = 1+1+1+1+1+1+1+1

1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525 — the number of partitions of n, A000041 at the Online Encyclopedia of Integer Sequences

And there are fractals — self-similarities at smaller and smaller scales — hidden in that simple arithmetic. Take the partitions of 8:

8 = 1+7 = 2+6 = 3+5 = 4+4 = 1+1+6 = 1+2+5 = 1+3+4 = 2+2+4 = 2+3+3 = 1+1+1+5 = 1+1+2+4 = 1+1+3+3 = 1+2+2+3 = 2+2+2+2 = 1+1+1+1+4 = 1+1+1+2+3 = 1+1+2+2+2 = 1+1+1+1+1+3 = 1+1+1+1+2+2 = 1+1+1+1+1+1+2 = 1+1+1+1+1+1+1+1 (c=22)

By definition, the sum of each partition of 8 is the same: 8. But the products — the result of multiplying the numbers of a partition — rise and fall wildly, hitting a maximum of 18 and a minimum of 1:

1.7 = 7
2.6 = 12
3.5 = 15
4.4 = 16
1.1.6 = 6
1.2.5 = 10
1.3.4 = 12
2.2.4 = 16
2.3.3 = 18
1.1.1.5 = 5
1.1.2.4 = 8
1.1.3.3 = 9
1.2.2.3 = 12
2.2.2.2 = 16
1.1.1.1.4 = 4
1.1.1.2.3 = 6
1.1.2.2.2 = 8
1.1.1.1.1.3 = 3
1.1.1.1.2.2 = 4
1.1.1.1.1.1.2 = 2
1.1.1.1.1.1.1.1 = 1

It’s interesting to ask when partitions(n) yield the biggest product (the answer is here). It’s also interesting to create graphs of prod(part(n)), the products of the partitions of n. You’ll see something I call partition fission. The graphs start to fissure into what look like fins or sails, and then each fin or sail starts to fissure too:

Graph for multiples of partitions(8) (partcount(8) = 22)

Graph for prod(part(9)) (partcount = 30)

prod(part(10)) (partcount = 42)

prod(part(11)) (partcount = 56)

prod(part(12)) (partcount = 77)

prod(part(13)) (partcount = 101)

prod(part(14)) (partcount = 135)

prod(part(15)) (partcount = 176)

prod(part(16)) (partcount = 231)

Those graphs are all on the same scale. The two graphs below have been adjusted to capture many more partitions and show the fractality coming into full flower:

prod(part(20)) (partcount = 627)

prod(part(28)) (partcount = 3718)

Finally, here’s an animated gif of the graphs for the partition-products of 8 to 16:

Animated gif for prod(part(8..16)) (animation at EZgif) (click for larger image)

Κοινόκοσμος καὶ Κωματόκοσμοι

by in Ancient Greek, Consciousness, Language, Literature, Philosophy, Quotations and tagged , , , , , , , | Leave a comment

• ὁ Ἡράκλειτός φησι τοῖς ἐγρηγορόσιν ἕνα καὶ κοινὸν κόσμον εἶναι τῶν δὲ κοιμωμένων ἕκαστον εἰς ἴδιον ἀποστρέφεσθαι. — Σέξτος Ἐμπειρικός, Πρὸς μαθηματικούς

• • Heraclitus said that for the waking is one common world, but the sleeping turn aside each into a world of his own. — Sextus Empiricus, fl. 150 A.D., Adversus Mathematicos / Against the Mathematicians (or: Against the Professors), VII. 129

Post-Performative Post-Scriptum

Κοινόκοσμος, Koinokosmos, “common-world”, “shared cosmos” ← κοινός, koinós, “common”, “shared” + κόσμος, kosmos, “world”, “order”, “universe”; καὶ, kai, “and”; Κωματόκοσμοι, Kōmatokosmoi, “sleep-worlds” ← Greek κῶμα, kôma, “deep sleep” + κόσμος, kosmos

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.

The Power of Cow

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

1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964, 3914488, 5736961, 8407925, …

• Narayana’s cows sequence: a(0) = a(1) = a(2) = 1; thereafter a(n) = a(n-1) + a(n-3). […] Number of digits in A061582. — A000930 at the Online Encyclopedia of Integer Sequences

1, 3, 9, 27, 621, 1863, 324189, 961232427, 2718369612621, 6213249182718361863, 1863961227324621324918324189, 32418927183662196121863961227324961232427, 961232427621324918186327183632418927183662196122718369612621, …

• a(1) = 1, a(n) = number obtained by replacing each digit of a(n-1) with three times its value. — A061582 at the OEIS

Vowel-Voided Verse

by in Language, Literature, Poetry and tagged , , , , , , , | Leave a comment

THE RUSSO-TURKISH WAR

WAR harms all ranks, all arts, all crafts appal;
At Mars’ harsh blast arch, rampart, altar fall!
Ah! hard as adamant a braggart Czar
Arms vassal-swarms, and fans a fatal war!
Rampant at that bad call, a Vandal band
Harass, and harm, and ransack Wallach-land.
A Tartar phalanx Balkan’s scarp hath past,
And Allah’s standard falls, alas! at last.

THE FALL OF EVE

EVE, Eden’s empress, needs defended be;
The Serpent greets her when she seeks the tree.
Serene she sees the speckled tempter creep;
Gentle he seems — perverted schemer deep —
Yet endless pretexts, ever fresh, prefers,
Vervetts her senses, revers when she errs.
Sneers when she weeps, regrets, repents she fell,
Then, deep-revenged, reseeks the nether Hell!

THE APPROACH OF EVENING

IDLING I sit in this mild twilight dim.
Whilst birds, in wild swift vigils, circling skim.
Light wings in sighing sink, till, rising bright.
Night’s Virgin Pilgrim swims in vivid light.

INCONTROVERTIBLE FACTS

NO monk too good to rob, or cog, or plot.
No fool so gross to bolt Scotch collops hot.
From Donjon tops no Oronooko rolls.
Logwood, not lotos, floods Oporto’s bowls.
Troops of old tosspots oft to sot consort.
Box tops our schoolboys, too, do flog for sport.
No cool monsoons blow oft on Oxford dons,
Orthodox, jog-trot, book-worm Solomons!
Bold Ostrogoths of ghosts no horror show.
On London shop-fronts no hop-blossoms grow.
To crocks of gold no Dodo looks for food.
On soft cloth footstools no old fox doth brood.
Long storm-tost sloops forlorn do work to port.
Rooks do not roost on spoons, nor woodcocks snort.
Nor dog on snowdrop or on coltsfoot rolls.
Nor common frog concocts long protocols.

PHILOSOPHY

DULL humdrum murmurs lull, but hubbub stuns.
Lucullus snuffs up musk, mundungus shuns.
Puss purrs, buds burst, bucks butt, luck turns up trumps;
But full cups, hurtful, spur up unjust thumps.

• from Literary Frivolities, Fancies, Follies and Frolics compiled by by William T. Dobson (1880)

Third Whirled Warp

by in Geometry, Mathematics, Squares, Triangles and tagged , , | Leave a comment

Here’s a regular hexagon inside a regular triangle, that is, an equilateral triangle:

Regular hexagon inside regular triangle

Imagine that two points are moving around the perimeter of each polygon, with the hex-point moving half as fast as the tri-point (after adjustment for the incommensurate relative lengths of the perimeters). If you trace the midpoint of the twin spinning points, you get this shape:

v3v6, 1 : 1/2, pol

And if you adjust the midpoint path as though the triangle had been stretched into a circle, you get this shape:

v3v6, 1 : 1/2, circ, pol

Here’s the same when the ratio of speeds is 1/2 to 1/3, that is, 1 to 2/3:

v3v6, 1/2 : 1/3, circ, pol

Without the polygons, it looks like this:

v3v6, 1/2 : 1/3, circ

When the ratio of speeds if -1/3 to 2/3, that is, the tri-point is moving counter-clockwise around the triangle, you get this shape:

v3v6, -1/3 : 2/3, pol

When it’s stretched into a circle, you get this:

v3v6, -1/3 : 2/3, circ, pol

It looks like a moustache:

v3v6, -1/3 : 2/3, circ

Here are more midpoint shapes created with a hexagon inside a triangle:

v3v6, 2/2 : 3/3, circ

v3v6, -1/2 : 3/4, circ

v3v6, 1/4 : 1/5, circ

v3v6, -1/4 : 3/4, circ

v3v6, -1/4 : 4/5, circ

v3v6, 2/3 : 3/4, circ

v3v6, 2/3 : 3/5, circ

v3v6, 3/4 : 4/5, circ

v3v6, 3/4 : 4/5, circ

Now try aligning the nested hexagon like this, so that the sides of the hexagon coincide with the middle third of the sides of the triangle:

v3v6, side alignment

With two points moving in a ratio of 1/3 to 1/4, you get this midpoint shape:

v3v6, sided, 1/3 : 1/4, pol

Here it is without the polygons:

v3v6, sided, 1/3 : 1/4

Now try a regular octagon inside a square:

v4v8, 1/2 : 1/3, circ, pol

v4v8, 1/2 : 1/3, circ

v4v8, -1/3 : 3/4, circ

v4v8, 2/3 : 3/5, circ

Now place a triangle inside a hexagon:

v6v3, 1 : 1/4, pol

If you stretch the midpoint path according to perimeter of the triangle, you get this:

v6v3, 1 : 1/4, circ, pol

v6v3, 1 : 1/4, circ

The three stretching shapes remind me of hands in Egyptian art, like this image of King Tutankhamun and Queen Ankhesenamun:

Detail from the Golden Throne of Tutankhamnun

Here are more midpoint paths:

v6v3, 1 : -1/4, circ

v6v3, 1 : 1/2, circ

v6v3, 1 : 1/3, circ

v6v3, -1 : 1/3, circ

v6v3, -1 : 1/4, circ

v6v3, 1 : 1/5, circ

v6v3, 2/3 : 1/4, circ

Now try a square inside an octagon:

v8v4, 2/3 : 1/4, circ, pol

v8v4, 2/3 : 1/4, circ

v8v4, 2/5 : 1/6, circ

v8v4, 2/5 : 3/7, circ

v8v4, 4/5 : 3/7, circ

Elsewhere Other-Accessible…

• First Whirled Warp — an earlier look at this kind of geometry
• Second Whirled Warp — and another earlier look