just something to say


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Guess – Compute consequences – Compare to experiment and see if it works.
(Richard Feynman)

One of those not necessarily useful things that go through your head when you wonder how the hell a hypercube weighs.

Imagining the dimensionless point means identifying a mere position on the plane. Then, ideally dragging the point on the same plane until it derives a straight line segment consisting of only length is a convention. That is, a mathematical artifice useful to represent ideal geometric figures in the real world.

So that, for example, the volume of a unit cube, of edge 1, is constructed by dragging the “starting” point into the three known dimensions. This is to say that originating from a point that is a simple location in space (or in the plane) it is possible to derive a segment devoid of thickness and qualified as a distance between two points, that is, consisting of only the virtual length and subsequently obtaining a square, equally lacking in thickness, side n.

Thus, in classical geometry it is imagined to “pull” a dimensionless entity, the point, to obtain a geometric entity (line or segment) defined by the only numerically quantified length n, to then eventually obtain a square of conventional surface n2 on the plane and finally a cube of volume n3 in space.

The consistency on the plane and subsequently in the space of the figures thus outlined (point, segment, square) is purely virtual. This is because it is not comprehensible or calculable the mass of a point that has no dimensions, as well as a line consisting of only the length. Nor that of a square with only two dimensions, which consists of a mere extension on the plane.

These figures are therefore abstract and massless. And consequently the quantity of matter (mass/weight) of a mathematical derivate three-dimensional cube would be just as evanescent.

It is difficult, in fact, defining the mass of the three-dimensional figure thus idealized with a volume, which came out of nothing but numbers. We should ask ourselves, first, how much a virtual point and a single-dimensional line weigh, and then what is the weight of the infinite square virtual surfaces necessary to compose an imaginary cube. (TAB A)

So, let’s first invent a heavy, material point, that is, a macroscopic quantum, whose minimum mass is unity. A three-dimensional object, that, for the sake of calculation, we choose to be a cubic shape with edge 1 and mass 1. Which also goes well with the granularity of nature.

From our starting “point” (that we therefore see as a small cube weighing one kilo), we can then build an object, we will fancy two-dimensional, but supplied with a mass, which we will still call segment. Visually it will be consisting of the sum of two contiguous cubes and it will be a small parallelepiped of thickness 1, length 2 and mass 2.

From the segment as indicated above, we can then derive a square of thickness 1, surface n2 (4) and corresponding mass m2 (4), and finally a cube of mathematical volume n3 (8) in the three-dimensional world.

Assuming that the mass of our object (i.e. our future hypercube) cannot be that of the figures ontologically belonging to the three-dimensional space only (where the volume n3 corresponds to the mass m3), we must instead imagine that its nature is the one suitable for its destination beyond the third dimension, since it is understandable that not only the hypercube we want to reach is extended in four or more dimensions, but also the starting cubes (quanta), which belong from the beginning to the world in four or five dimensions.

We therefore assume that the masses of the units that will form the object of our experiment (intended to be measured in the fourth and subsequent dimensions) must be consistent – starting from its very minimum components – to the world of four or more dimensions. (TAB B)

In other words, a fourth or fifth or subsequent dimension, either exists or not. And if it’s there (exactly what we assume), there is no reason why the unit cube, our starting “point”, as well as the other transit figures leading to the hypercube, are not, first, extended and therefore massive as suggested by the nature and the rules of the objects of the fourth/fifth, or subsequent dimension. Entities invisible to us and hardly imaginable, but not immeasurable for what concerns their mass.

We then mathematically launch our object into the fourth dimension (n4) and imaginatively call quantity (something like an hyper-volume) its geometric measure in the fourth dimensional world (the one which we cannot see). Then, we weigh the quanta/mass deriving from the subsequent calculations and observe the increase in volume, quantity and mass on the path from the second, to the third and then to the fourth dimension.

To the sample cube deriving from the square of side 2, which will become quantity n4 (16) (i.e. sixteen units in the shape of the fourth dimension) is subsequent a hypercube which has mass n8 (256) units.

The exponential progression activated considering the relationship between size and mass of this object, thus shows us that to a cube “n+1”, deriving from a square of side 3, inasmuch as it is already projected in the fifth dimension, corresponds a measure of 81 units of quantity and a mass of n8 (6,561) units in the 4th dimension. And by the next step up to the fifth dimension we will obtain an invisible virtual quantity of 35 (243) units and a mass of 59,046 units. (TAB C)

And here we stop because it is peculiar the thought of our invisible and massive five-dimensional object/event compared to the meters and kilos of the part that we can perceive.

The newly conceived five-dimensional monster, of which however as beings bound to the three-dimensional world, we cannot see and measure anything else than the appearance and/or intersection (Edwin A. Abbot, Flatland) of a cube of three units of edge and 27 miserable kilos of mass, would weigh, instead, given its nature, in terrestrial conditions (gravity), at rest, over 59 tons.

I don’t know if it’s dark, but it looks like a lot of matter. A real shame not to be able to see how it is made.


Sample cube made of eight unit cubes (re. TAB B)

 Cube “n+1” (re. TAB C) (Rubik’s Cube)

Tesseract (Never mind, I’m here because I’m cute)

Written by pipistro

January 13, 2021 at 7:46 pm

Posted in Uncategorized

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