Hilbert's Grand Hotel

 

Hilbert's Grand Hotel, 8" x 10", linocut by Ele Willoughby, 2021

My latest linocut print illustrates the paradox of Hilbert's Grand Hotel, a thought experiment conceived by the great mathematician David Hilbert (1862-1943) in 1924 to show the paradox of infinite sets. If you imagine a regular hotel with a finite number of rooms, if they are all filled you can't add more guests. In an imaginary infinite hotel however, guests are always welcome! Even if all the (conveniently numbered 1, 2, 3, ...) rooms are filled, you simply shuffle all the guests over one room and make room for the next guest. You can repeat this argument for one more guest, two more guests, three more guests, ...infinity more guest... infinity groups of infinite guests, etcetera!

The print shows a grand hotel on a steep angle (with a vanishing point at infinity of course), labelled "Hilbert's Grand Hotel" below a cameo of Hilbert himself, and the slogan "No vancancy. Check in now!" Above the hotel is a glowing infinity symbol. The size of countably infinite sets, like all natural numbers {1,2,3, .... ∞ } is denoted by Aleph Nought ℵ0 shown on the two flags.

COSMIC turtles

A graphic cosmic timeline, from the
Big Bang to the present day
(Credit: NASA/WMAP Science Team)

This year, to coincide with PrintAustin, Art.Science.Gallery is hosting COSMIC, a show of prints about "Origin of the Universe. Evolution of the Universe. String Theory. Dark Matter. Dark Energy. Multiverse. Unification of Space + Time. Our Solar System. Cultural Cosmology." I mean, how could I resist? I've been working in every spare moment to produce two brand new prints for the show. (As it happens, most of my astronomy related prints have appeared in previous exhibits there, so I needed new cosmological material).

My first instinct, the first images I imagined were something like the NASA spacetime diagram shown, which is a graphical representation of a lot of the major concepts of cosmology: the Big Bang, Inflation, the formation of galaxies and so forth and even dark energy and our current accelerated expansion... It's a great use of imagery to communicate science, but it's far too literal. It's a diagram, not a piece of art. So I turned to the figurative, imaginative, metaphorical, to avoid the trap of the literal.

Turtles, All the Way Down, 11" x 14" (27.9 cm x 35.6 cm)
linocut with chine-collé by Ele Willoughby, 2016

In some cultures there are origin myths explaining that the (apparently flat) Earth is in fact supported on the back of the World Turtle (occassionally in conjunction with other beasts). In cosmology - or the astrophysics of the origins of our Universe - there is an expression "turtles all the way down" which relates to a well-known anecdote, and metaphor for the problem of infinite regress. There are many versions of the anecdote. It appears, famously, in Stephen Hawking's A Brief History of Time, amongst other sources. The story goes that a physicist, after explaining the origins of our planet, is confronted by a disgruntled audience member who claims this is nonsense. The Earth is, according to him or her, supported by a great turtle. When the physicist asks about what supports the turtle, the answer is a larger turtle, of course. When the physicist persists and asks what supports this next turtle, why, it's "turtles, all the way down!" of, course. The same problem in epistemology is known as the Münchhausen trilemma, after after the story of Baron Munchausen who claimed to have pulled himself and his horse out of the mire by pulling his own hair, in the ultimate example of bootstrapping. (Incidentally, bootstrapping is a favorite tool of physicists, and The Adventures of Baron Munchausen, is one of my favorite films).

A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: 'What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise.' The scientist gave a superior smile before replying, 'What is the tortoise standing on?' 'You're very clever, young man, very clever,' said the old lady. 'But it's turtles all the way down!'

— Hawking, A Brief History of Time, 1988

I've illustrated this expression with a variety of wonderful turtle species using collaged or chine-collé papers to show their various colours. From the top we have an eastern mud turtle (Kinosternon subrubrum), a red-eared slider (Trachemys scripta elegans), a black pond turtle (Geoclemys hamiltonii), a painted batagur (Batagur borneoensis), a Oaxaca mud turtle (Kinosternon oaxacae), a green sea turtle (Chelonia mydas) and the hint of something larger.

Those who've known me a long time will know that I've illustrated this concept before, I love turtles, am obsessed with origin myths and am rather found of zaratans. I am now wondering what, if anything, is the difference between a zaratan and an aspidochelone (incidentally, also encounted by none other than the Baron Munchausen, of course)?

The second piece I've submitted for COSMIC is Noh Spacetime. So it's metaphors all the way down.

Ele Willoughby, 'Noh Spacetime', linocut 12" x 12", 2015

Schrödinger's cat

Is Schrödinger's cat in the box, or not? It depends on when you look at this linocut! This colour-changing thermochromic block print shows the famous thought-experiment of renown quantum physicist Erwin Schrödinger (who would never hurt a real cat!). Both the cat in blue and the poison in pink will disappear when the print is exposed to heat.

In struggling to fully explain the strangeness of the quantum world, which can only be described in terms of probabilities and wave functions, Schrödinger suggested a sort of metaphor, at the size of every day things - the scale of classical physics as we know it. He imagined a cat in a steel box with a vial of poison which might be opened if, and only if, a radioactive decay occurs. In one half-life of the radioactive material, there is a 50:50 chance that the material has decayed or not. So, if the box is closed, and we cannot see within, we can only describe the state of the cat in terms of probabilistic wave functions. After one half-life, we would be forced to describe the contents of the box as the sum of the half likelihood of a live cat and the half likelihood of a dead cat. It is as if, to the outside world, there exists both a live and dead cat.... until, one opens the box and looks. Then we know we either have not yet had the radioactive decay and subsequent release of poison, so the cat is fine, or the radioactive material has decayed and the cat is no more. So, what's so special about looking in the box? Does the wave function "collapse" onto one of these two possibilities? Does the universe split into one in which the cat lives and one in which the cat does not, as in the many-worlds interpretation of quantum mechanics? I tend to side with Niels Bohr whose Copenhagen Interpretation of quantum mechanics says, as all experimentalists know, unless you observe something you cannot determine its state. However, physics cannot answer this question! We can determine the probabilities only, we cannot say why. But that's okay. Paradox is delicious.


This is one print in an edition of 10. Each print is made with the box and vial printed in normal, silver block printing ink. The cat and poison are printed using thermochromic powder and block printing medium so when heated above 30°C (86 F) then they turn colourless and disappear. Thus, as in the thought experiment, you don't know whether there is a cat in the box without looking at the print. Each print is 12.3 by 12.5 inches (31.2 cm by 31.8 cm) and made on Japanese kozo (or mulberry) paper.

This print is one of a series of 'Imaginary Friends of Science', including Maxwell's Demon, Laplace's Demon and Descartes' Demon. It is also one of a collection of thermochromic prints which include a portrait of Louis Pasteur (with disappearing bacteria) and a Groundhog (with disappearing shadow).

As much as I would like to do another portrait of the one and only Minouette, I couldn't let her model for this one, even if it is only a thought experiment.