The 6-fold way

A fabulous design from Elizabeth Daingerfield Zwicky yesterday:

(#1) “6-fold” (or: “The 6-fold Way”)

To come. On 6-fold symmetry: snowflakes (natural and in paper), many monocot flowers, Kekulé’s carbon ring for benzene, the major colors of the color wheel (reproduced in the rainbow flag for Gay Pride).

Then on number, color, and gender parallelisms, which will give us 6 as purple and queer. And how the opposition of the secondary hues green with purple in #1 parallels the opposition of the primary hues blue with red (and, in the background of #1, the opposition of the primary hues red with yellow).

And on the name 6-fold way, adapted from the Noble Eightfold Path of Buddhism and Murray Gell-Mann’s adaptation of the idea (under the name The Eightfold Way) to a theory organizing subatomic particles.

In sixes. In nature, 6-fold symmetry is famously seen in snowflakes:

(#2) A snow crystal

And in art, in paper snowflakes like this one:


(Gut-easy schemes for cutting paper snowflakes give you 8-fold symmetry. To get 6, you need to make a fold into 3. It’s important here that 6 = 2 x 3 (the product of the two smallest prime numbers, what you might think of as the basic primes).)

In the plant world, 6-petaled flowers are common, especially in monocots (daffodils, crocuses, tulips, lilies, daylilies):

(#4) Stargazer lily: 6 petals, in 2 sets of 3 (2 x 3 again)

In general, monocots have 3 or 6 petals. (The three-petaled trillium even has threeness built into its name.)

(Other common petal numbers are 4 — in the crucifers, which get their family name from their cross-like flowers — and 5 — throughout the rose family, from strawberry plants through almond trees.)

On to invisible nature, in the form of molecular structures. From my 4/19/13 posting “B.C. portmanteau”, Kekulé’s carbon ring structure for benzene (one of the great scientific ideas of the 19th century):

(#5) 2 x 3: 3 sets of single bonds, 3 of double bonds

Then the color wheel, in its simple version, a set of primary hues (blue, red, yellow) and one of secondary hues (green, purple, orange):

(#6) 2 x 3: 3 primaries, 3 secondaries

These six colors are reproduced, in linear order (red at top, purple at bottom), in the now-standard rainbow Pride flag.

Parallelisms. The numbers 2 and 3 are associated by synesthetes with various colors, but especially blue and red, respectively:

2 ‖ blue ∷  3 ‖ red

(with ‖ ‘is parallel to’ and ∷ the ‘as’ of proportion)

From Wikipedia:

Grapheme-color synaesthesia or colored grapheme synesthesia is a form of synesthesia in which an individual’s perception of numerals and letters is associated with the experience of colors.

Then colors have gender associations. Again, these relationships are complex and variable, but the association of saturated blue with masculinity and saturated red with femininity is fairly strong:

blue ‖ masculine ∷  red ‖ feminine

Chaining these, we get:

2 ‖ blue ‖ masculine ∷  3 ‖ red ‖ feminine

Then, using the fact that 6 is a compound of 2 and 3, we expect its associated color to be a compound of blue and red, that is purple, and its associated gender to be a compound of masculine and feminine, that is, queer:

∴ 6 ‖ purple ‖ queer

So 6 is a queer number.

(I use compound with some thought, not specifying the exact nature of the compounding operation (which is obviously different in the different domains). Queerfolk are sometimes said to be “two-spirited” or to be both male and female, but these interpretations aren’t necessary.)

In any case, the idea that purple is a queer color because it’s a combination of masculine blue and feminine red is a natural one.

Then there are the colors in #1, where the main figure is a study in green (G) and purple (P), not blue (B) and red (R). But there’s a way to analogize from B → R to other relationships within the color wheel: move counterclockwise with the hue class (primary or secondary), as here:

(#7) (Apologies for the wretched artwork, which is mine)


green : purple ∷ blue : red

so that green and purple reproduce the roles of blue and red, respectively. And in the background:

red : yellow ∷ blue : red

so that red and yellow reproduce the roles of blue and red.

You can see it all as a shifted version of the blue-red relationship. In 6 = 2 x 3 parts.

The 8-fold Way. The source of my ttle “The 6-fold Way” in #1. It starts in Buddhism. From Wikipedia:

(#8) The Noble Eightfold Path, in a diagram

The Noble Eightfold Path (Pali: ariyo aṭṭhaṅgiko maggo, Sanskrit: āryāṣṭāṅgamārga) is an early summary of the path of Buddhist practices leading to liberation from samsara, the painful cycle of rebirth.

… The Eightfold Path teaches that by restraining oneself, cultivating discipline, and practicing mindfulness and meditation, house-leavers (monks and nuns) attain nirvana and stop their craving, clinging and karmic accumulations, thereby ending their rebirth and suffering.

(#9) The eight-spoke Dharma wheel symbolizes the Noble Eightfold Path

And then to physics, with hexagon + 2. From Wikipedia:

(#10) The meson octet. Particles along the same horizontal line share the same strangeness, s, while those on the same diagonals share the same charge, q.

In physics, the Eightfold Way is a theory organizing subatomic hadrons. The name was coined by American physicist Murray Gell-Mann as an allusion to the Noble Eightfold Path of Buddhism. It led to the development of the quark model. An equivalent theory was independently proposed by Israeli physicist Yuval Ne’eman.

You’ll need to go back to sources to understand the details. Meanwhile, go back and enjoy #1 some more.


One Response to “The 6-fold way”

  1. arnold zwicky Says:

    May 20th: another wonderful 6-fold from EDZ, quite different in tone from the one above:

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