I have often found myself, in the past, struggling to contemplate “tetradimensional” space.  Of course, this is by design of the human mind, highlighted so well in the classic story “Flatland“:

The story is about a two-dimensional world referred to as Flatland which is occupied by geometric figures, line-segments (females) and regular polygons with various numbers of sides. The narrator is a humble Square, a member of the social caste of gentlemen and professionals in a society of geometric figures, who guides us through some of the implications of life in two dimensions. The square has a dream about a visit to a one-dimensional world (Lineland) which is inhabited by “lustrous points.” He attempts to convince the realm’s ignorant monarch of a second dimension but finds that it is essentially impossible to make him see outside of his eternally straight line.

The narrator is then visited by a three-dimensional sphere, which he cannot comprehend until he sees Spaceland for himself. This sphere, who remains nameless, visits Flatland at the turn of each millennium to introduce a new apostle to the idea of a third dimension in the hopes of eventually educating the population of Flatland of the existence of Spaceland. From the safety of Spaceland, they are able to observe the leaders of Flatland secretly acknowledging the existence of the sphere and prescribing the silencing of anyone found preaching the truth of Spaceland and the third dimension. After this proclamation is made, many witnesses are massacred or imprisoned (according to caste).

After the Square’s mind is opened to new dimensions, he tries to convince the Sphere of the theoretical possibility of the existence of a fourth (and fifth, and sixth …) spatial dimension. Offended by this presumption and incapable of comprehending other dimensions, the Sphere returns his student to Flatland in disgrace.

He then has a dream in which the Sphere visits him again, this time to introduce him to Pointland. The point (sole inhabitant, monarch, and universe in one) perceives any attempt at communicating with him as simply being a thought originating in his own mind (cf. Solipsism):

‘You see,’ said my Teacher, ‘how little your words have done. So far as the Monarch understand them at all, he accepts them as his own – for he cannot conceive of any other except himself – and plumes himself upon the variety of Its Thought as an instance of creative Power. Let us leave this God of Pointland to the ignorant fruition of his omnipresence and omniscience: nothing that you or I can do can rescue him from his self-satisfaction.’[3]
— the Sphere

The Square recognizes the connection between the ignorance of the monarchs of Pointland and Lineland with his own (and the Sphere’s) previous ignorance of the existence of other, higher dimensions.

Once returned to Flatland, the Square finds it difficult to convince anyone of Spaceland’s existence, especially after official decrees are announced – anyone preaching the lies of three dimensions will be imprisoned (or executed, depending on caste). Eventually the Square himself is imprisoned for just this reason.

It is just difficult for a mind to think in a “higher dimension”.  Of course, terms like “higher dimension” are usually indicative that the subject at hand is utter nonsense, religious babble.  However, for the purpose of this article the term applies to adding different spatial dimensions, as indicated in the Flatland excerpt above.

The normal 3 dimensions are relatively easy to understand:  left/right, up/down, forward/backward.  It is the normal 3 dimensions that we deal with every day, despite being trapped on the ground and generally only interacting with a 2 dimensional mindset because of this.

But to add in a 4th spatial dimension can give a headache (it has me, anyway).  To contemplate a 4th spatial dimension, I will use the simple concept of “in/out” (although nothing traps it into a “in/out” paradigm, other than our own imagination).  A good graphic of this would be the following animated tesseract:

If you watch this image, you can clearly see that the cube is moving from “in” to “out”.

Of course, realizing this just adds in more questions, such as what is physically in a direction of “in”?  Is this where gravity travels towards?

Regardless of the additional questions, being able to visualize tetradimensional space is the “in” thing to do.


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