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Tekel-Mirim – Tolkien's Polyhedra

A Visit to Tekel-Mirim

The following is an extract from:

      J.R.R. Tolkien, "The Notion Club Papers", Sauron Defeated, HarperCollins, 1992, pp 207-9

This manuscript was Tolkien's second unfinished attempt at a novel involving time travel from our own modern world to his land of Numenor in the Lord of the Rings mythology. He never did finish it. Tekel-Mirim is a world set in a universe which overtly seeks to harmonise Tolkien's fantasy universe with the science-fictional worlds of his old friend and fellow Inkling C S Lewis. Here, a member of the future (to Tolkien) Notion Club has in his dreams been travelling to other worlds across both space and time.

‘I thought that as I wandered there I came to a little world, of our Earth’s size more or less ... Tekel-Mirim it was, a land of crystals.

‘Whether the crystals were really of such great size – the greatest were like the Egyptian pyramids – it is hard to say. Once away from Earth it is not easy to judge such things without at least your body to refer to. For there is no scale; and what you do, I suppose, is to focus your attention, up or down, according to what aspect you wish to note. And so it is with speed. Anyway, there on Tekel-Mirim it was the inanimate matter, as we should say, that was moving and growing: into countless crystalline formations. Whether what I took for the air of the planet was really air, or water, or some other liquid, I am not able to say; though perhaps the dimming of sun and stars suggests that it was not air. I may have been on the floor of a wide shallow sea, cool and still. And there I could observe what was going on: to me absorbingly interesting.

‘Pyramids and polyhedrons of manifold forms and symmetries were growing like ... like geometric mushrooms, and growing from simplicity to complexity; from single beauty amalgamating into architectural harmonies of countless facets and reflected lights. And the speed of growth seemed very swift. On the summit of some tower of conjoined solids a great steeple, like a spike of greenish ice, would shoot out: it was not there and then it was there; and hardly was it set before it was encrusted with spikelets in bristling lines of many pale colours. In places forms were achieved like snowflakes under a microscope, but enormously larger: tall as trees some were. In other places there were forms severe, majestic, vast and simple.

‘For a time I could not count I watched the “matter” on Tekel-Mirim working out its harmonies of inherent design with speed and precision, spreading, interlocking, towering, on facet and angle building frets and arabesques and frosted laces, jewels on which arrows of pale fire glanced and splintered. But there was a limit to growth, to building and annexation. Suddenly disintegration would set in – no, not that, but reversal: it was not ugly or regrettable. A whole epic of construction would recede, going back through shapeliness, by stages as beautiful as those through which it had grown, but wholly different, till it ceased. Indeed it was difficult to choose whether to fix one's attention on some marvellous evolution, or some graceful devolving into – nothing visible.

‘Only part of the matter on Tekel-Mirim was doing these things (for “doing” seems our only word for it): the matter that was specially endowed; a scientist would say (I suppose) that was of a certain chemical nature and condition. There were floors, and walls, and mighty circles of smooth cliff, valleys and vast abysses, that did not change their shape nor move. Time stood still for them, and for the crystals waxed and waned.

‘I don't know why I visited this strange scene, for awake l have never studied crystallography, not even though the vision of Tekel-Mirim has often suggested that I should. Whether things go in Tekel-Mirim exactly as they do here, I cannot say. All the same I wonder still what on earth or in the universe can be meant by saying, as was said a hundred years ago (by Huxley, l believe) that a crystal is a “symmetrical solid shape assumed spontaneously by lifeless matter". The free will of the lifeless is a dark saying. But it may have some meaning: who can tell? For we have little understanding of either term. I leave it there. l merely record, or try to record, the events I saw, and they were too marvellous while I could see them in far Tekel-Mirim for speculation.’


‘In Tekel-Mirim I must have been not only far away in Space but in a time somewhat before my earth-time.’


Planetary Analogue
Penitentes in the Atacama desert (via Wikimedia Commons)

Tolkien's vision may have been closer to reality than he knew. Penitentes are long, jagged growths of ice that rear upwards in very cold, dry and windy places where snow and ice lie semi-permanently on the ground. The combined action of sunlight and wind can cause them to grow to considerable heights. In Antarctica they can reach several metres and on Jupiter's moon Europa they are estimated to reach 15 m tall.

Given a change in the weather, a penitente that had slowly grown its crystalline way to adulthood could suddenly start to shrink back down again, until the weather changed once more.

Tolkien's methodical vagueness about both the physical size and timescale of his crystal growths make it easy to imagine some world on which penitentes could behave as he describes. His "great steeple, like a spike of greenish ice", his forms "like snowflakes under a microscope, but enormously larger" and "frosted laces" are all redolent of the same water ice.

One is left wondering whether Tolkien had ever read an account of these extraordinary growths, written by some Antarctic explorer.

Polyhedra and J W Dunne

Tolkien's imaginary travels through space and time were inspired in large part, as were those of C S Lewis, by the writings of J W Dunne, who experienced precognitive dreams and in 1927 published a theory of time and consciousness in his 1927 book An Experiment with Time to try and explain them. Tolkien and Lewis both owned copies of the bestselling 3rd edition.

Dunne's theory makes another unlikely, and somewhat eerie, appearance in the literature of polyhedra. During his lifetime, geometers began exploring higher dimensions of space and discovering higher-dimensional analogues of polyhedra which they called polytopes. Paul J. Donchian was an American of Armenian descent, whose ancestors had been jewellers. Around 1925 he too had a series of precognitive dreams, this time of such a nature that, despite not being a mathematician, he determined to understand the geometry of higher dimensions. He undertook a series of delicate and beautiful models of polytopes projected into ordinary space and presented them in public exhibitions. He became a good friend of the famous geometer H S M "Donald" Coxeter and photographs of some of them found their way into Coxeter's most well-known work, Regular Polytopes, first published in 1963.

In the same work, Coxeter criticised Dunne for mistakenly treating Einstein's spacetime as Euclidean. Dunne does makes this association at one point (The Serial Universe, 1st Edn 1934, p.144) but his context is not clear. The remark cannot be taken in isolation, as his next paragraph discusses Minkowski's use of √−1 (the square root of minus one) as a time multiplier. There was at that time some debate as to whether this feature was inherent in spacetime or was just an artefact of Einstein's equations; indeed, the point remains mildly vexed to this day. Dunne's ignorance can (on this occasion at least) be forgiven.

For more about these crystalline mathematical forms, see Guy's Polyhedra Pages

Updated 5 Dec 2018