Le Ventre de Paris

The Belly of Paris must always have been a place worth visiting. After the food market was dismantled, Les Halles became a gigantic shopping center. I have not seen it since the new construction began a few years ago.

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In any case, the area is a place worth visiting without wallet. At some places, we cannot tell anymore whether we are inside or outside.

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Architecture permeates everything, even the layout of the boutiques. The lady was not pleased with me taking the picture and called security. And this was in 1991.

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Long passageways in almost black and white made me think of Alain Resnais.

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Escaped, one wonders if Henri de Miller’s sculpture L’Écoute in front of the nearby church of St Eustace ever gets a quiet moment.

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In Memoriam

There is nothing like Paris. Before and after a backpacking vacation in the French Alps in 1991, I spent a few days just walking around in the city.

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To honor the city and its people, I have scanned and edited the negatives from these walks, as a personal work of memory.
The view above is from the Centre Georges Pompidou. The spooky sky is caused by shooting through the plexiglass windows surrounding the outside escalators of the building.

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The French have a wonderful tradition how their presidents invest enormous sums in art and culture.

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Right outside, the Stravinsky Fountain, with sculptures by Jean Tinguely and Niki de Saint Phalle, vibrant with colors and life.

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Then there is the Arab World Institute, one of the Grands Projets of François Mitterrand.

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Another project Mitterrand completed: The conversion of a railway station into a museum, the Musée d’Orsay.

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This walk will continue.

Costaesque (Algorithmic Geometry V)

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In 1982, Celso José da Costa wrote down the equations of a minimal surface that most mathematicians at that time thought shouldn’t exist. It shares properties with the plane and catenoid that were supposed to be unique to them. Nothing could be more wrong. Since Costa, many more minimal surfaces in that same elite class have been found.

Costa
The curiously complicated way in which the Costa surfaces merges a horizontal plane with a catenoid by avoiding any intersections has become a pattern for similar constructions that is quite aptly called Costaesque.

Amusingly, the same pattern occurs in Alan Schoen’s I6 or Figure Eight surface from 1970.

SchoenI6

It can be viewed as a triply periodic aunt of the Costa surface but was conceived as a Plateau solution for two pairs of squares in parallel planes, each of which meet a corner to form a figure eight.

This surface has a particularly simple polygonal approximation by the bent 60 degree rhombi that we have encountered before.

I6b

Let’s take 12 such bent rhombi and assemble them into an X-piece that has the two figure 8 squarical holes. A second such X-piece is rotated by 90 degrees and attached to the top to form the polygonal version of Schoen’s I6 fundamental piece.

I6a

Alternatively, one can also tile the surface with straight 60 degree rhombi so that it becomes a triply periodic zonohedron.

I6c

The Grand Gulch

Spring Break 1994 took me to Utah. After 24 hours in the car the landscape started to look like Max Ernst would have painted it.

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The entire week we (a group of eight members of CHAOS) would spend hiking through a large part of the Grand Gulch, a primitive area in the south eastern corner of Utah.

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This meant packing food for six days, and hoping that there would be enough water.

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Hiking through a canyon like this can be claustrophobic. After descending to the canyon floor, one is constantly surrounded by unclimbable walls, and the barren vegetation is little consolation.

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But of course the landscape is full of surprises, with new views at every turn. And then there are the Anasazi ruins.

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The Ancestral Puebloans (or Anasazi) were a large Native American civilization that disappeared after 1150 CE, likely due to a climate change. Not much is known about them, but in the Grand Gulch one can find their cliff dwellings and pictograms.
There are worse things to leave behind.

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Neovius surface (Algorithmic Geometry IV)

When the truncated octahedron tiles space, the diagonals of the hexagonal faces become part of a line configuration.

TruncatedOctahedronTiling

Following these lines we can build the bent rhombi that we encountered in Schwarz’ P-surface, but here we will focus on the more complicated bent octagons that weave around the square faces of the truncated octahedra. These serve as Plateau contours for another minimal surface, the Neovius surface, named after the Finnish mathematician Edvard Rudolf Neovius, a student of Hermann Amandus Schwarz.

Neoviusminimal

One can also fill each octagon with four copies of said bent rhombus to obtain an interesting polygonal version of the Neovius surface. Here are two such filled octagons aligned. Note that we have broken a rule: The four bent rhombi that fill the octagon are not rotated about their edges to fit together, but reflected.

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Rotating about the edges by 180 degrees will create larger portions of the infinite surface.

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Temporarily breaking a rule can sometimes be a good thing.

Neoviuspoly

Water in Motion (Iceland XIV)

The law of gravity is still intact.

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Whenever in doubt, contemplating a healthy waterfall is certain remedy.

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The imposed free fall gives direction, determination and diverts the attention from situations where indecision has become a permanent state.

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So, is that it? Do we have to either submit to a higher power, or be tossed around by pure chance?

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Sometimes, for a few seconds, this koan has an answer.

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