Symmetrical Geometrical Objects

I have been repeatedly referring to the (infinite) reciprocal sums of the unique number sequences associated with (x – k)ⁿ = 0 and especially (x – 1)ⁿ = 0.

And I refer to these reciprocal sums associated with (x – 1)ⁿ = 0 as the Alternative Zeta 2 function i.e. Alt ζ₂(n).

However it is possible now to provide a more general formula.

This is based on the fact that the nth term of these unique number sequences follow a definite pattern.

Then, for (x – 1)², the unique number sequence is 1, 2, 3, 4 with the kth = k and the

                            _{∞                                                                  ∞}

sum of  terms  = ∑ k/1! with (infinite) sum of reciprocals = ∑1!/k.    

                           ^{k=1                                                                n=1} 

For (x – 1)³, the unique number sequence is 1, 3, 6, 10, … with the kth term = k(k + 1)/2!

                                   _{∞                                                                       ∞}

And sum of terms  = ∑k(k + 1)/2! with sum of reciprocals = ∑ 2!/{k(k + 1)}.

                                  ^{k=1                                                                     k=1} 

Then, giving one more example to illustrate a consistent pattern, for (x – 1)⁴, the unique number sequence is 1, 4, 10, 20, …, with kth term = k(k + 1)(k + 2)/3!   

                                                             _∞

So the (infinite) sum of reciprocals = ∑ 3!/{k(k + 1)(k + 2)}.

                                                            ^k=1

Therefore for the general case (x – 1)ⁿ = 0, the (infinite) sum of reciprocals of the associated unique number sequences of this simple polynomial equation i.e. Alt ζ₂(n), is given as

_∞

∑ (k – 1)!/{k(k + 1)(k + 2) … (k + n – 2)}.

^k=1

Note, however that the Alt ζ₂(n) function remains undefined for n = 1!

A fascinating geometrical way of looking at the unique number sequences given by

(x – 1)ⁿ = 0, is through number patterns that are associated with the simplest perfectly symmetrical objects that can be constructed in the various dimensions.

For example the simplest symmetrical polygon figure (with least number of sides) that can be constructed in 2-dimensional space is a 3-sided equilateral triangle.

The diagram in Mathworld then illustrates how the triangular numbers - which correspond to the unique number sequences associated with (x – 1)³ = 0 - then arise.

So we start with 1 dot and with the construction of an equilateral triangle we then place a dot at each vertex = 3. Then by placing a dot midway along each line, the total no. of points = 6.

And then with 4 dots equally spaced out along each line (with another placed within the triangle), when the total no. = 10 (overall) we can preserve the perfect symmetry of equal distance between adjacent dots. 

Then in the final example we place 5 dots equally spaced on each side of the triangle with 3 more internally = 15 (overall). 

What is also interesting here is that we can join up the dots in each triangle to form a number of smaller triangles that represents the previous number in the series.

Thus in the simplest case in the 3 dot case we can join the points to form 1 triangle (which represents the previous number in the sequence). Then in the case of the 6 dot triangle we can join the dots to form 3 upright triangles (with same orientation to the 1 larger triangle contained). Then in the case of the 10 dot triangle we can form 6 upright triangles and in the 15 dot triangle 10 upright triangles of same orientation to larger triangle and so on.

Then the simplest 3-dimensional figure that is perfectly symmetrical with respect to its angular rotations is the tetrahedron which has the equilateral triangle as its base.

Though harder to represent (in 2-dimensional) terms, it is easy enough to see how the number of symmetrically arranged points (associated with this 4-sided figure follow the unique number sequence associated with (x – 1)⁴ = 0, i.e. 1, 4, 10, 20, …

So once again we start with 1 dot represented by the red dot at the top of the diagram (from Mathworld). Then to picture the simplest symmetrical tetrahedral figure we combine the 3 blue dots (representing the base of a tetrahedron with the red dot above) to obtain 4.

Then at the next level where each line of the base of the tetrahedron is divided in 2 we have now 6 green dots which combine the 3 blue and 1 red above to complete the new tetrahedron with 10 dots. Then with the next larger tetrahedron, we have 10 brown points at the base combined with the 6, 3 and 1 respectively above to complete the tetrahedron with 20 points.

So we can easily see here how this sequence of dots, representing the fully symmetrical nature of the 3-dimensional tetrahedron (with the least number of sides i.e. 4, possible in 3-dimensional space) is the unique digit sequence for (x – 1)⁴ = 0, i.e. 1, 4, 10, 20, 35, … 

However what is perhaps remarkable is that these insights can then be extended for the simplest fully symmetrical objects in n-dimensional space.

Thus - though impossible to properly visualise in 3 or less dimensions - we can equally envisage for example a 4-dimensional polytope, as a 5-sided equivalent in 4 space dimensions to the 4-sided tetrahedron in 3 space. This is known as a 5-cell or tetrahedral pyramid. (See Wikipedia entry).

Thus once again this contains the least number of sides i.e. 5, that a fully symmetrical object can possess in 4 space dimensions.  

So what we can now say therefore is that the unique number sequence associated with

(x – 1)⁵ = 0, i.e. 1, 5, 15, 35, 70, …  now describes the appropriate number of equally spaced points with respect to this object (as the number of points on each side progressively increases). 

Thus in general terms, the unique sequence of numbers associated with the polynomial equation
(x – 1)ⁿ = 0, encodes the manner, in which the equally spaced points of the simplex n-sided fully symmetrical geometrical objects, occurs in (n – 1) space dimensions.