Elementary Anagram

An exceptionally ingenious anagram by the American mathematician Mike Keith (born 1955):

hydrogen + zirconium + tin + oxygen + rhenium + platinum + tellurium + terbium + nobelium + chromium + iron + cobalt + carbon + aluminum + ruthenium + silicon + ytterbium + hafnium + sodium + selenium + cerium + manganese + osmium + uranium + nickel + praseodymium + erbium + vanadium + thallium + plutonium

iiiiiiiiiiiiiiiiiiiiiiiiiii + uuuuuuuuuuuuuuuuuuuuuuuuuu + mmmmmmmmmmmmmmmmmmmmmmmmmm + nnnnnnnnnnnnnnnnnnnn + eeeeeeeeeeeeeeee + rrrrrrrrrrrrrr + ooooooooooooo + llllllllllll + aaaaaaaaaaaa + tttttttttt + ccccccc + hhhhhh + bbbbbb + ssssss + dddd + yyyy + ggg + ppp + z + f + v + x + k

nitrogen + zinc + rhodium + helium + argon + neptunium + beryllium + bromine + lutetium + boron + calcium + thorium + niobium + lanthanum + mercury + fluorine + bismuth + actinium + silver + cesium + neodymium + magnesium + xenon + samarium + scandium + europium + berkelium + palladium + antimony + thulium

[as chemical names]

1 + 40 + 50 + 8 + 75 + 78 + 52 + 65 + 102 + 24 + 26 + 27 + 6 + 13 + 44 + 14 + 70 + 72 + 11 + 34 + 58 + 25 + 76 + 92 + 28 + 59 + 68 + 23 + 81 + 94




7 + 30 + 45 + 2 + 18 + 93 + 4 + 35 + 71 + 5 + 20 + 90 + 41 + 57 + 80 + 9 + 83 + 89 + 47 + 55 + 60 + 12 + 54 + 62 + 21 + 63 + 97 + 46 + 51 + 69

[as atomic numbers]

Elsewhere other-accessible…

Mike Keith — official website (with anagram here)

Tright Sights

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)


ვენერა — რიგით მეორე პლანეტა მზიდან; მისი სიმკვრივე და აგებულება მსგავსია დედამიწისა (Translate.ge)

金星 — 金 jīn, jìn, gold; metals in general; money; 星 xīng, a star, planet; any point of light (MDBG)

I Like Dyke

Grumpy Dyke’s Phallocrator (2023)

• “Stoner rock meets industrial metal in the suppurating bowels of Hell.” — Garth Halloran
• “See ’em live or die!” — Jessica Tővetz
• “A rancidly toxic maxi-melange of racism, sexism, transphobia, open support for MAGA and open hostility to BLM. But Grumpy Dyke have an unacceptable side too.” — The Quietus

Elsewhere other-accessible…

Grumpy Dyke at Bandcamp


• R I O A E T L U

→ Herriot a été élu.

• L N N E O P Y I A V Q E I E D C D

→ Hélène est née au pay grec, y a vécu et y est décédée.

• J J A D I D A C K O T L A H E T D B K C D G A L E V D I N C P I E D F I J E C O Q P D B B A J T

→ Gigi a des idées assez cahotées: elle a acheté des bécasses et des geais, a élevé des hyènes, s’est payé des effigies et s’est occupée des bébés agités.

• From John Julius Norwich’s More Christmas Crackers (1990)

Extra Tetra

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).

Wander in Woods

Errantes silva in magna et sub luce maligna
inter harundineasque comas gravidumque papaver
et tacitos sine labe lacus, sine murmure rivos,
quorum per ripas nebuloso lumine marcent
fleti, olim regum et puerorum nomina, flores.

Cupido Cruciatur, Decimius Magnus Ausonius (c.310-c.395)

They wander in deep woods, in mournful light,
Amid long reeds and drowsy headed poppies,
and lakes where no wave laps, and voiceless streams,
Upon whose banks in the dim light grow old
Flowers that were once bewailed names of kings.

• translated by Helen Waddell in her Medieval Latin Lyrics (1929)

First Whirled Warp

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.

Previously pre-posted (please peruse)

We Can Circ It Out — more on converting polygons into circles

Light on Kite

kite, n.

Forms:  Old English cyta, Middle English ketekijtkuytte, Middle English kuyte, Middle English–1600s kyte, (1500s kightkightekyghtScottish kyt), Middle English kite.

Etymology: Old English cýta ( < *kūtjon-); no related word appears in the cognate languages.

1. A bird of prey of the family Falconidæ and subfamily Milvinæ, having long wings, tail usually forked, and no tooth in the bill.

2. [ < its hovering in the air like the bird.] A toy consisting of a light frame, usually of wood, with paper or other light thin material stretched upon it; mostly in the form of an isosceles triangle with a circular arc as base, or a quadrilateral symmetrical about the longer diagonal; constructed (usually with a tail of some kind for the purpose of balancing it) to be flown in a strong wind by means of a long string attached. Also, a modification of the toy kite designed to support a man in the air or to form part of an unpowered flying machine. — Oxford English Dictionary

Krøyers Øje: Sand og Strand

Sommeraften på Skagen Sønderstrand (1893), Peder Severin Krøyer (1851-1909)

(click for larger)

Elsewhere other-accessible…

Summer Evening on Skagen’s Southern Beach — more about the painting