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Imagine losing or gaining a spatial dimension

Living in a 3D world, my mind was pleasingly warped when I watched this 7-minute , explaining what a 4D ball would look like in 3D. I sent it to Apatt who likened it to And He Built a Crooked House, which in turn, reminded me of a book I鈥檇 heard of, Flatland.

Reading them one after the other was enjoyably challenging. This is a review, and star rating, of both.

And He Built a Crooked House, by Robert Heinlein

This is an early short story (1941) by a big name in sci-fi. It's about a 4D construction.

Quintus Teal is a young architect who thinks of a house as 鈥�a machine for living, a vital process, a live dynamic thing鈥�. His big idea is to use the fourth spatial dimension to 鈥�put an eight-room house on the land now occupied by a one-room house. Like a tesseract鈥�. He goes on to explain a lot of geometry, and I would have been bemused if I hadn鈥檛 already seen illustrations of what a tesseract, aka hypercube, would look like.

We鈥檙e all familiar with a and its net:


Then add a dimension for a and its net:


For more of an explanation, see video links at the end of this review.

Teal achieves his dream 鈥�by using strong girders and folding money鈥� and takes his friends, Mr and Mrs Bailey to see it.
鈥�That's the grand feature about a tesseract house, complete outside exposure for every room, yet every wall serves two rooms and an eight-room house requires only a one-room foundation.

But four dimensions don鈥檛 fit easily into a three dimensional world, and things become strange. Stranger than any staircases or buildings that drew.

Flatland, by Edwin A Abbott

About a 2D world and published in 1884.

This short book uses geometry as an analogy for a socio-political satire of Victorian attitudes to class and social mobility; gender (in)equality; biological determinism, evolution, and eugenics; religion and the supernatural. It probably works better for those with more knowledge of and interest in the period than I have. Fortunately, my edition had lots of notes, and Abbot included several illustrations.

The key concept is that in a two-dimensional world, a three-dimensional figure is observed as a cross-section. If it's moving, it is seen as a series of cross-sections, and can appear to appear and disappear out of nowhere.

In the first half, the narrator, A Square, explains the very hierarchical social system of the two-dimensional Flatland. The more sides a shape has, and the more regular it is, the higher up the scale it is. 鈥�Delicate females鈥� are are mere lines, 鈥�wholly devoid of brain-power, and have neither reflection, judgement, nor forethought, and hardly any memory鈥�, but Abbot himself was a priest and headmaster who believed in women鈥檚 education, including at university.

The second half has more of a story: 鈥�my initiation into the mysteries of Space鈥� and 鈥�the Gospel of Three Dimensions鈥�. A Square develops a theoretical and then practical understanding of other spatial dimensions. The trouble is, that鈥檚 unimaginable and heretical in Flatland.

The progression of understanding of different spatial dimensions is well explained, with a few illustrations: A square is 鈥�a Line of Lines鈥� and a sphere is 鈥�many Circles in one鈥�. However, Abbott鈥檚 love of initial capital letters was a little distracting, and in part of the second section the language becomes overly Biblical and deferential (thee, thou, My Lord etc).


Image: 鈥淭he Weeping Woman鈥�, in which 鈥淧icasso used cubist forms of fragmentation to depict the face in a series of angular planes鈥�. ()

Clearly explained information about 2D and 4D

To understand the fourth dimension, you need a firm grasp of the zeroth, first, second, and third dimensions. Unless you have a mathematical background, I strongly suggest you watch one or two of these videos before reading either story. I found it helpful to see the same ideas explained in slightly different ways.

鈥� The fourth dimension explained, via second and third, in six minutes, .

鈥� Hypercubes explained and illustrated, without sound, in five minutes, .

鈥� Five-minute TED animation, summarising the story of Flatland and explaining 2D and 4D, .

鈥� Animation of the story of Flatland, narrated by Carl Sagan, in four minutes, .

鈥� Animation of the story of Flatland, made in 1965, eleven minutes, .

鈥� Carl Sagan explains Flatland and goes on to the fourth dimension, in nine minutes, .

The two stories

鈥� You can read Flatland, free, in a variety of digital formats, on Gutenberg, .

鈥� You also can find the text of And He Built a Crooked House online.


Image: 鈥淢r Osborne, may I be excused? My brain is full.鈥� (The Far Side, by Gary Larson)

See also

鈥� If it鈥檚 all too much, there鈥檚 Norton Juster鈥檚 delightful picture book, The Dot and the Line, which I reviewed HERE. I included illustrations and a link to an Oscar-winning animation.

鈥� Malcolm wrote two short stories about very high-tech homes in The Martian Chronicles, which I reviewed HERE. Usher II is a subversive dystopian comedy that's also a tribute to Poe's Fall of the House of Usher. There Will Come Soft Rains starts off almost slapstick, but takes a very tragic turn.