And every poem and every picture
a sensation in the eye and heart
Something that jolts you awake
from the rapt sleep of living
in a flash of pure epiphany
where all stands still
in a diamond light
for what it truly is
in all its mystery
So a bird is an animal
flown into a tree
singing inscrutable melodies
As a lover stands transparent
screened against the sun
smiling darkly in the blinding light
Poem #46 from A Far Rockaway Of the Heart, 1997.
Lawrence Ferlinghetti March 24, 1919 – February 22, 2021
A polystick is a connected finite subgraph of the grid graph, and a tetrastick is a polystick with four edges. There are 25 of them, counting mirror copies.
In other words, these are squiggles you can make with four strokes. They’d make a nice alphabet for people who are addicted to abstraction.
Today we are focussing on six of them, fattened and colored above. They are denoted by the letters H, O, F, +, and the mirrors of H and F. For reasons to become clear later we consider O and + also as mirrors of each other in a certain sense. The goal is to tile rectangles with them, like in the 3×7 and 2×12 rectangles below.
There are many constraints on what tiles one can use, and how many. For instance, an a x b rectangular grid has a(b+1) vertical and (a+1)b horizontal edges, for a total of 2ab+a+b edges. This number is divisible by 4 only if a-b is divisible by 4, so squares are good for tiling, as are the two rectangles above. They both consist of 52 segments and thus require 13 tetrasticks. Below is a different example.
Note that all our 6 letter except for H and its mirror use two horizontal and two vertical segments. As the 3×7 rectangle has 4 more horizontal edges than vertical ones, we need at least four H-tetrasticks (or its mirrors) to tile this rectangle. We can use more, but then only an even number of them. Likewise, we need at least two H-tetrasticks to tile the 2×12 rectangle.
This brings us to today’s puzzle: Tile the 3×7 rectangle with your choice of 13 tetrasticks from our selection of six, and then use the same set to tile the 2×12 rectangle. The examples on this page are attempts that require to flip an H or an F into its mirror (or an O into +). Can you find a perfect solution that doesn’t require flipping a tetrastick over?
The ice on the lake shone so brightly that it did not look like ice at all.
Frozen into this block of ice were broad, sword-shaped leaves, thin straws, seeds and detritus from the woods, a brown, straddling ant – all mingled with bubbles that had formed and which appeared clearly as beads when the sun’s rays reached them.
And what was this? It must be the ice palace.
But this was unexpected, too: she was standing in what looked like a room of tears.
No one is involved deeply enough to be present. A blast of noiseless chaos may cause the air to vibrate in distant bedrooms, but no one wakes up to ask: What is it? No one knows. Now the palace, with all its secrets, goes into the water-fall. There is a violent struggle, and then it has gone.
Quotes from The Ice Palace by Tarjej Vesaas, translated by Elizabeth Rokkan.