# Time Pyramid

Manfred Laber’s public art piece in Wemding, Germany, doesn’t look much like a pyramid yet. That’s because a new concrete block is laid only every 10 years; the structure was begun in 1993 and will be completed in the year 3183, when the 120th block is placed at the top.

Altogether that’s 1,200 years, the town’s age when Laber conceived the project and laid its foundation.

# Black and White

By Otto Georg Edgar Dehler. White to mate in two moves.

# Turán’s Brick Factory Problem

During World War II, Hungarian mathematician Pál Turán was forced to work in a brick factory. His job was to push a wagonload of bricks along a track from a kiln to storage site. The factory contained several kilns and storage sites, with tracks criss-crossing the floor among them. Turán found it difficult to push the wagon across a track crossing, and in his mind he began to consider how the factory might be redesigned to minimize these crossings.

After the war, Turán mentioned the problem in talks in Poland, and mathematicians Kazimierz Zarankiewicz and Kazimierz Urbanik both took it up. They showed that it’s always possible to complete the layout as shown above, with the kilns along one axis and the storage sites along the other, each group arranged as evenly as possible around the origin, with the tracks running as straight lines between each possible pair. The number of crossings, then, is

$\displaystyle \mathrm{cr}\left ( K_{m,n} \right ) \leq \left \lfloor \frac{n}{2} \right \rfloor \left \lfloor \frac{n-1}{2} \right \rfloor \left \lfloor \frac{m}{2} \right \rfloor \left \lfloor \frac{m-1}{2} \right \rfloor ,$

where m and n are the number of kilns and storage sites and $\displaystyle \left \lfloor \right \rfloor$ denotes the floor function, which just means that we take the greatest integer less than the value in brackets. In the case of 4 kilns and 7 storage sites, that gives us

$\displaystyle \left \lfloor \frac{7}{2} \right \rfloor \left \lfloor \frac{7-1}{2} \right \rfloor \left \lfloor \frac{4}{2} \right \rfloor \left \lfloor \frac{4-1}{2} \right \rfloor = 18 ,$

which is the number of crossings in the diagram above.

Is that the best we can do? No one knows. Zarankiewicz and Urbanik thought that their formula gave the fewest possible crossings, but their proof was found to be erroneous 11 years later. Whether a factory can be designed whose layout contains fewer crossings remains an open problem.

# Werner’s Nomenclature of Colours

Today it’s possible to describe a color quantitatively, but how did people make such fine distinctions in the 18th century? German geologist Abraham Gottlob Werner proposed a solution in 1774: His Von den äußerlichen Kennzeichen der Foßilien included a “color dictionary” that located each hue in the natural world. Updated by Scottish painter Patrick Syme, it describes 110 colors, telling where each might be found in animal, vegetable, and mineral form: Number 35, for example, “bluish lilac purple,” is the shade of the male of the dragonfly Libellula depressa, the blue lilac, and the mineral lepidolite. Number 82, “tile red,” may be found in the breast of the cock bullfinch, in the shrubby pimpernel, and in porcelain jasper.

This common language gave naturalists an objective way to communicate what they were seeing. Off Brazil aboard the H.M.S. Beagle in 1832, Charles Darwin wrote, “I had been struck by the beautiful color of the sea when seen through the chinks of a straw hat. It was according to Werner nomenclature ‘Indigo with a little azure blue’. The sky at the time was ‘Berlin [blue] with little Ultra marine’.”

The Internet Archive has Syme’s full text.

# Oops

In 1816 the United States built a fort on Lake Champlain to guard against attacks from British Canada.

Too late the planners discovered that they’d chosen a site north of the border — they’d built their fort in Canada.

It’s now called “Fort Blunder.”

# Poser

What’s the next letter in this sequence?

OTTFFSSEN

# Lessons Learned

Aphorisms of Lazarus Long, the 2,000-year-old protagonist of Robert A. Heinlein’s 1973 novel Time Enough for Love:

• Always store beer in a dark place.
• Small change can often be found under seat cushions.
• If you don’t like yourself, you can’t like other people.
• It’s amazing how much “mature wisdom” resembles being too tired.
• Certainly the game is rigged. Don’t let that stop you; if you don’t bet, you can’t win.
• Get a shot off fast. This upsets him long enough to let you make your second shot perfect.
• The truth of a proposition has nothing to do with its credibility. And vice versa.
• A brute kills for pleasure. A fool kills from hate.
• It may be better to be a live jackal than a dead lion, but it is better still to be a live lion. And usually easier.
• If it can’t be expressed in figures, it is not science; it is opinion.
• Your enemy is never a villain in his own eyes. Keep this in mind; it may offer a way to make him your friend. If not, you can kill him without hate — and quickly.
• Cheops Law: Nothing ever gets built on schedule or within budget.
• No state has an inherent right to survive through conscript troops and, in the long run, no state ever has. Roman matrons used to say to their sons: “Come back with your shield, or on it.” Later on, this custom declined. So did Rome.
• Never appeal to a man’s “better nature.” He may not have one. Invoking self-interest gives you more leverage.
• By the data to date, there is only one animal in the Galaxy dangerous to man — man himself. So he must supply his own indispensable competition. He has no enemy to help him.
• A zygote is a gamete’s way of producing more gametes. This may be the purpose of the universe.

And “A generation which ignores history has no past — and no future.”

# “Mutual Problem”

Said Jerome K. Jerome to Ford Madox Ford,
“There’s something, old boy, that I’ve always abhorred:
When people address me and call me ‘Jerome’,
Are they being standoffish, or too much at home?”
Said Ford, “I agree; it’s the same thing with me.”

— William Cole

# Noted

The angle cos-1(-1/3) = 109.47°, familiar from soap films and tetrahedral molecular geometry, can be produced with an ordinary piece of A4 paper: Because it has a width:length ratio of $1:\sqrt{2}$, folding it corner to corner as shown yields a shape with precisely that angle.

(Nick Lord, “A ‘Maths Bite’: How to Impress a Chemist,” Mathematical Gazette 80:489 [1996], 584-584.)