# See-Saw Jaw

From Sierpiński triangle to T-square to Mandibles (and back again) (animated)
(Open in new window if distorted)

Elsewhere other-accessible…

Mandibular Metamorphosis — explaining the animation above
Agnathous Analysis — more on the Sierpiński triangle and T-square fractal

# Hextra Texture

A hexagon can be divided into six equilateral triangles. An equilateral triangle can be divided into a hexagon and three more equilateral triangles. These simple rules, applied again and again, can be used to create fractals, or shapes that echo themselves on smaller and smaller scales.

# Fractal Fourmulas

A square can be divided into four right triangles. A right triangle can be divided into a square and two more right triangles. These simple rules, applied again and again, can be used to create fractals, or shapes that echo themselves on smaller and smaller scales.

# Go with the Floe

Fractals are shapes that contain copies of themselves on smaller and smaller scales. There are many of them in nature: ferns, trees, frost-flowers, ice-floes, clouds and lungs, for example. Fractals are also easy to create on a computer, because you all need do is take a single rule and repeat it at smaller and smaller scales. One of the simplest fractals follows this rule:

1. Take a line of length l and find the midpoint.
2. Erect a new line of length l x lm on the midpoint at right angles.
3. Repeat with each of the four new lines (i.e., the two halves of the original line and the two sides of the line erected at right angles).

When lm = 1/3, the fractal looks like this:

(Please open image in a new window if it fails to animate)

When lm = 1/2, the fractal is less interesting:

But you can adjust rule 2 like this:

2. Erect a new line of length l x lm x lm1 on the midpoint at right angles.

When lm1 = 1, 0.99, 0.98, 0.97…, this is what happens:

The fractals resemble frost-flowers on a windowpane or ice-floes on a bay or lake. You can randomize the adjustments and angles to make the resemblance even stronger:

Ice floes (see Owen Kanzler)

Frost on window (see Kenneth G. Libbrecht)

# O Apollo

One of Swinburne’s most powerful, but least-known, poems is “The Last Oracle”, from Poems and Ballads, Second Series (1878). A song in honour of the god Apollo, it begins in lamentation:

Years have risen and fallen in darkness or in twilight,
Ages waxed and waned that knew not thee nor thine,
While the world sought light by night and sought not thy light,
Since the sad last pilgrim left thy dark mid shrine.
Dark the shrine and dumb the fount of song thence welling,
Save for words more sad than tears of blood, that said:
Tell the king, on earth has fallen the glorious dwelling,
And the watersprings that spake are quenched and dead.
Not a cell is left the God, no roof, no cover
In his hand the prophet laurel flowers no more.

And ends in exultation:

For thy kingdom is past not away,
Nor thy power from the place thereof hurled;
Out of heaven they shall cast not the day,
They shall cast not out song from the world.
By the song and the light they give
We know thy works that they live;
With the gift thou hast given us of speech
We praise, we adore, we beseech,
We arise at thy bidding and follow,
We cry to thee, answer, appear,
O father of all of us, Paian, Apollo,
Destroyer and healer, hear! (“The Last Oracle”)

The power, grandeur and beauty of this poem remind me of the music of Beethoven. Swinburne is also, on a smaller scale and in a different medium, one of the geniuses of European art. He and Beethoven were both touched by Apollo, but Apollo was more than the god of music and poetry: he also presided mathematics. But then mathematics is much more visible, or audible, in music and poetry than it is in other arts. Rhythm, harmony, scansion, melody and rhyme are mathematical concepts. Music is built of notes, poetry of stresses and rhymes, and the rules governing them are easier to formalize than those governing, say, sculpture or prose.  Nor do poetry and music have to make sense or convey explicit meaning like other arts. That’s why I think a shape like this is closer to poetry or music than it is to painting:

(Image from Wikipedia.)

This shape has formal structure and beauty, but it has no explicit meaning. Its name has a divine echo: the Apollonian gasket or net, named after the Greek mathematician Apollonius of Perga (c.262 BC–c.190 BC), who was named after Apollo, god of music, poetry and mathematics. The Apollonian gasket is a fractal, but the version above is not as fractal as it could be. I wondered what it would look like if, like fleas preying on fleas, circles appeared inside circles, gaskets within gaskets. I haven’t managed to program the shape properly yet, but here is my first effort at an Intra-Apollonian gasket:

(If the image does not animate or looks distorted, please try opening it in a new window)

When the circles are solid, they remind me of ice-floes inside ice-floes:

Simpler gaskets can be interesting too:

# Tri Again

All roads lead to Rome, so the old saying goes. But you may get your feet wet, so try the Sierpiński triangle instead. This fractal is named after the Polish mathematician Wacław Sierpiński (1882-1969) and quite a few roads lead there too. You can create it by deleting, jumping or bending, inter alia. Here is method #1:

Divide an equilateral triangle into four, remove the central triangle, do the same to the new triangles.

Here is method #2:

Pick a corner at random, jump half-way towards it, mark the spot, repeat.

And here is method #3:

Bend a straight line into a hump consisting of three straight lines, then repeat with each new line.

Each method can be varied to create new fractals. Method #3, which is also known as the arrowhead fractal, depends on the orientation of the additional humps, as you can see from the animated gif above. There are eight, or 2 x 2 x 2, ways of varying these three orientations, so eight fractals can be produced if the same combination of orientations is kept at each stage, like this (some are mirror images — if an animated gif doesn’t work, please open it in a new window):

If different combinations are allowed at different stages, the number of different fractals gets much bigger:

• Continuing viewing Tri Again.