About Flatland and, on how to construct a friend of dimension 4 for A. Square
In 1800, Charles H. Hinton -the epitome of the fever for the geometrical fourth dimension who coined the term Tesseract to refer to the 4-D hypercube- wrote an essay called “What is the Four Dimension?”. Hinton’s work inspired Abbott’s masterpiece “Flatland: A Romance of Many Dimensions”, published in 1884.
Flatland, a parody of the Victorian era, is A. Square’s story, the story of the Prometheus of the land of -apparently- dimension 2. A. Square was a two dimensional (plus some kind of epsilon >0 height) mathematician who lived in a highly stratified, misogynist and rigid society regulated by the Aristocratic Constitution of Flatland States. In Flatland, geometry was destiny: both regularity and the number of sides determined the social position of the 2-D entity.
The greater the number of sides, the higher the social status of the Flatlander: the nobility was composed by regular n-gons called Circles (thousands of sides), the priests. Squares and Pentagons belonged to the professional class. Equilateral triangles were part of the middle class and Isoceles were associated to the lowest class, the soldiers. Women (rectangles of infinitesimal width, conceived as Straight Lines) were almost above criminals: irregular plane figures in a place where a “connection between geometrical and moral irregularity” was perceived as the result of an immutable, Natural Law.
There were three methods of recognition between Flatlanders: followed by the hearing method (voice-discrimination was a plebeian virtue), the feeling method was meant for both Women and the lower classes: -Sir., can I feel you?, then the Felt was supposed to stand perfectly still not to cause, with his angles, an irreparable injury to the Feeler. Upper classes were trained on recognition by sight, an Aristocratic skill acquired through Aristocratic education.
Attend to your configuration (doesn’t it sound a little bit like Plato?): not an iota of egalitarianism was free of suppression. Long ago, after a bloody rebellion, Pantocyclus, the Chief Circle of those days, prohibited color. To talk about a higher dimension -evidently- would be analogous to call into question the superiority Circles. It was crystal clear: persons pretending to have received revelations from another World would be condemned to death or imprisonment.
The third Millennium was about to start, it was the last day of the 1999th when the Sphere made an appearance:
3³ has an obvious Geometrical meaning… I am indeed, in a certain sense a Circle … I am many Circles in one…
I am not a plane Figure, but a Solid. You call me a Circle; but in reality I am not a Circle, but an infinite number of circles, of size varying from a Point to a Circle of thirteen inches in diameter, one placed on the top of the other.
How hard it was for A. Square to imagine a third dimension? I challenge you to think of an hyperbeing, an object of four dimensions. The daring effort of the Sphere to make A. Square understand his inherent property, his spatial nature, included both, the “Analogy Argument” and a brief journey through Spaceland:
Tell me, Mr. Mathematician; if a point moves Northward and leaves a luminous wake, what name would you give to that wake?… Now conceive the Northward straight Line moving parallel to itself East and West… Now stretch your imagination a little, and conceive a Square in Flatland, moving parallel to itself upward… out of Flatland altogether.
… a Point has 0 sides; a Line has 2 sides; a Square has 4 sides; 0, 2, 4, … what is the next number (of this arithmetical sequence)? Six.
Let’s distinguish spatial dimension from an object’s dimension. Intuitively, we can think of an object of dimension n living the best life inside an n dimensional space, right? That is because an extra dimension implies an extra degree of freedom for a point in space, that is why A. Square, an (n-1) dimensional object that can live in 𝔼ⁿ, cannot happily move in Spaceland. While the dimension (the Hamel dimension) of a Vector space is just the cardinality of the basis (the set of linearly independent vectors spanning the entire space), the dimension of an object can be lower than the space in which it is embedded.
The Analogy Argument is not but a glint of mathematical induction where the base case lives in Pointland. The dimension of topological space X is either the Menger-Urysohn (small) inductive dimension ind(X) or the large inductive dimension Ind(X). Let’s focus on ind(X) by first defining what does it mean for a point to be a boundary point.
Let A⊂ X and p ∈ A, p∈ ∂ A , where ∂ A is the boundary of A, iff every open ball containing p intersects both A and its complement Aᶜ. If the space consists of just one point p: X={p}, then the boundary of the space is empty, i.e.: ∂ X={} (can you see why?).
Now, ind({})=−1 and the dimension of a point is zero. ind(X)≤ n if ∀ p∈ X, ∃ V open, V⊂ X, p∈ V such that ind(∂ V)≤ n−1. If ind(X)≤ n and ind(X)>n, then ind(X)=n. In plane words, we can reveal the dimension of an object/space (in terms of open sets) by looking at its boundary: think of A. Square as the wrap of a chocolate called A. Sphere.
Deeds, not words. A. Sphere extracted A. Square from Flatland:
I looked, and behold, a new world! There stood before me, visibly incorporate all that I had before inferred, conjectured, dreamed, of perfect Circular beauty… a beautiful harmonious something… the surface of the Sphere…
Every mathematician knows that mathematical induction is equivalent to the well ordering principle. Thus, if Lineland is not just a dream and k-dimensional beings exist, we can use the inductive step:
My Lord, your own wisdom has taught me to aspire to One even more great, more beautiful, and more closely approximate to Perfection than yourself… some higher purer region… some yet more spacious Space, some more dimensionable Dimensionality… there is the Argument from Analogy of Figures…
In One Dimension, did not a moving Point produce a Line with TWO terminal points?
In Two Dimensions, did not a moving Line produce a Square with FOUR terminal points?
In Three Dimensions, did not a moving Square produce… that blessed Being, a Cube , with EIGHT terminal points?
And in Four Dimensions shall not a moving Cube… result in a still more divine Organization with SIXTEEN terminal points?… 2, 4, 8, 16 (,… , 2ⁿ): is not this a Geometrical Progression?… this other Space is really Thoughtland…
Unfortunately, when it comes to power, human nature is an invariant, no matter the dimension. A Sphere failed to withstand the fact: there are n kinds of hyperbeings living either in a higher dimensional physical reality or in Dedekinnd’s Gedankenwelt. Since n∈ℕ, A. sphere is infinitely inferior. It was the end of the trip.
A. Square, the apostle of the Gospel of the Three Dimensions ended up in jail. Why A. Sphere did not put his friend out of jail? We will never know.
… Prometheus up in Spaceland was bound for bringing down fire for mortals, but i -poor Flatland Pormetheus- lie here in prison for bringing down nothing to my countrymen.
Nonetheless, we can help A. Square by constructing, for him, a 4-D friend: a kind-hearted hypercube ready to remove him from his cell and to show him another example of higher dimensionality.
Let {0,1}ⁿ be the set of all n−bit strings. The n−dimensional hypercube is the graph Qₙ= (V_{Qₙ}, E_{Qₙ} ), where V is the set of nodes V={(a₁, a₂,…, aₙ): aⱼ ∈ {0,1}ⁿ, n ∈ ℕ }. Now, the node (a₁, a₂,…, aₙ) is adjacent to the node (b₁, b₂,…, bₙ) iff aⱼ=bⱼ ∀ j∈{1,2,…,n} except for just one bit. It is not hard to prove that the number of nodes of Qₙ is 2ⁿ and that the number of edges is equal to n2ⁿ⁻¹.
Using the definition of Qₙ, we can construct Q₁, Q₂ and Q₃ as follows:
Even if you have found the definition of Qₙ somehow confusing, could you notice the pattern? Fine, because we have constructed a handsome, blue-eyed hypercube:
Now, it is time to find A. Square. Should we construct a Time Machine?
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