The Mathematics of the Witch House, Part 3 – Higher Dimensions

Symbols from the Necronomicon (from, by Vemisery)

In a previous article on hyperspace I mentioned that  Michio Kaku describes in his book Hyperspace: A Scientific Odyssey Through Parallel Universe, Time Warps, and the Tenth Dimension that the laws of nature become simpler and more elegant when expressed in higher dimensions.  Since Euclid and up to the mid-19th century, flat or planar geometry has been THE only geometry of western civilization, used in various practical applications including architecture.  However, more so than presenting the subject of geometry in a cohesive format, it was Euclid’s methods of using logic, deductive reasoning, evidence and proofs that is his larger contribution to mathematics and science.

While Euclid’s contributions were monumental, they also limited alternative thought and theory in geometry and mathematics in general.  In planar geometry one dimension is a straight line ( a point has no dimensions), two dimensions is plane shape (square, circle, triangle, etc.) and three dimensions can be represented as solids (cubes, spheres, cylinders, etc.).  Each higher dimension is created by adding another line at a 90 degree angle and once you get to three dimensions there is no additional “directions” for another line.  Thus, the conclusion was there is no 4th dimension.

Representation of one, two and three dimensions (; Duane B. Karlin)

As I have previously mentioned many mathematicians and philosophers agreed with this.  For example, Ptolemy from Alexandria  stated that “proof that the fourth dimension is impossible.” – Michio Kaku, 1994.  This quote is very similar to some of the statements made in Robert Weinberg’s article H.P. Lovecraft and Pseudomathematics (Discovering H.P. Lovecraft by Darrell Schweitzer, 1995) such as “The existence of higher dimensions should be of little concern to this world as any contact with such dimensions is impossible.”, “There is no way that we can construct a four-dimensional object.”, and “It is impossible (not unlikely or not yet possible, but impossible, actually shown to be never possible) to construct a higher dimension from a lower one.”

I bring this quotes up not to criticize Mr. Weinberg but to make the point I get the impression that when he wrote this article in 1971 he was in the camp with many of the classical mathematicans that higher dimensions are impossible.

However, as a teenager of the 1980’s I got to watch the series and read the book Cosmos by Carl Sagan.  During one episode / chapter (The Edge of Forever) Sagan spoke about creatures living in flatland.  I strongly recommend you either rent the DVD of Cosmos, read the book or at least look up Sagan’s flatland discussion on YouTube.  The point I want to make here is that while we can’t “see” a 4th dimensional being with our three dimensional senses, just like a two dimensional being can not see a three dimensional being, experiments can be done to detect the presence of a higher dimension.

In Cosmos, Sagan made the point that a 3-dimenional “apple creature” passing through the 2-dimensional plan of Flatland (moving from above to under Flatland) would appear as a point, an expanding line, then a contracting line, then as another point and then disappear.  The square or triangle creature observing this may think they are insane and would not be able to understand what they just saw.  Could something similar happen if a 4th dimensional creature passed through our 3 dimensional world?  Is it possible that some individuals have actually experienced this, resulting in nervousness, phobia or even severe mental disorders such as schizophrenia?

Carl Sagan’s apple creature visiting Flatland

While we can’t see a 4th dimensional creature or object we may be able to see an interpretation or representation of it.  Again, from Carl Sagan’s Cosmos, you can make a representation of a 3-dimensional object (say a cube) on a 2-dimensional surface or plan (see below)

Point = 0D, Line = 1D, Square =2D, Cube =3D and Tesseract =4D

A three dimensional cube held by Carl Sagan

Every school child learns how to draw a representation of a cube on a 2 dimensional surface (piece of paper).   This same method can be done to draw a representation of a “hypercube” – a 4 dimensional cube.  Below is a “3-dimensional” representation of a hypercube, also known as a Tesseract.

A hypercube also known as a Tesseract

Again, this is just a representation of what a Tesseract looks like not a Tesseract itself, since we can’t look into the 4th dimension.  Thus, the strange shapes and images Walter Gilman sees and experiences he has as he travels through hyperspace – the prisms, labyrinths, limitless abysses of inexplicably coloured twilight, and bafflingly disordered sound; clusters of cubes and planes, groups of bubbles, octopi, centipedes, living Hindoo idols, and intricate arabesques roused into a kind of ophidian animation – may essentially be his 3-dimensional mind trying to perceive and comprehend higher dimensions.

To make one final point on Mr. Weinberg’s critical analysis of that story, I can’t actually see an electron but I can conceptually visualize what one looks like (a small particle, a traveling wave or hazy cloud) and more importantly I can see evidence of electrons (see below).  Now I understand that Mr. Weinberg’s article is dated (early 1970’s), Cosmos was in the 1980’s and most serious higher dimensional thought and research is less than 20 years old; this obviously has to be taken into account in this analysis.  So I can certainly understand stating that something is very unlikely but to say impossible may be a little too hasty (in my opinion).

Electron particle tracks (credit: Brookhaven National Laboratory / Science Photo Library).  Do these particle tracks somewhat resemble the curious symbols found in the Necronomicon?

Next time the discussion will concentrate on hyperspace travel in The Dreams in the Witch House and later articles will discuss Brown Jenkin and the appearance of Elder Things and the Black Man (Nyarlathotep).  Thank you.  Fred


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