The Set of Least Inclusive Geometric Units in N Dimensi= ons

&= nbsp; “Every Set has a least element.”

&= nbsp;

&= nbsp; We define the Least Inclusive Geometric Unit as that Geometric Unit existing i= n a given D which uses the least possible number of elements native to that exp= ression of D and yet exhibits the full expression of the dimensions contained within that D. It is the Least Eleme= nt of the Set of Geometric Units within that expression of D. For example, within D=3D0 the Poin= t is that geometric unit which fully expresses the potential of that dimension.<= span style=3D'mso-spacerun:yes'> No other geometric unit is possible within D=3D0, so the Point is by default also the Least Inclusive Geometric= Unit for D=3D0. <= o:p>

In D=3D1, the infinite line is a not a possible geometric unit, since the conc= ept of geometric unit must imply finiteness. Geometric units possible within D=3D1 are those expressed by consecu= tive line segments, defined by a series of points along the line. This series constitutes an infinit= e set; but there is a lesser possibility: the single line segment, defined by two separate points on the line. = For D=3D1 the Line Segment is the Least Inclusive Geometric Unit (LIGU).

In D=3D2, there are an infinite number of possible geometric units that can be= built by combining line segments; triangle, square, pentagon, etc. The geometric unit that expresses = all the possibilities of D=3D2, i.e. Area, using the least amount of resources = is the Triangle, and for D=3D2, the Triangle is the LIGU.

Likewise, for D=3D3 the LIGU is the Tetrahedron.&nbs= p; There are an infinite number of possible elements for this set, also. Note, we have not stipu= lated regularity as a requirement of the geometric units within a given expressio= n of D.

In the case of D=3D4 we are faced with a puzzle. What are the characteristics of th= e LIGU for that D? Much has been wri= tten about the Hypercube, and its geometry is well understood, but is it the LIGU for D=3D4? In order to answer= this question, we will examine the development of the Set of Least Inclusive Geometric Units in N Dimensions.

	Points	Line Seg	Triangle<= /span>	Tetrahedron	Hyper-Tet	????????<= /span>
D=3D0	1	0	0	0	0	0
D=3D1	2	1	0	0	0	0
D=3D2	3	3	1	0	0	0
D=3D3	4	6	4	1	0	0
D=3D4	5	10	10	5	1	0
D=3D5	6	15	20	15	6	1

We can see that as D increases, the number of Points in the LIGU is augmented = by one element for each increase in D. However, at first glance it does not appear that there is a simple function that describes the progress of the number of Line Segments, Triang= les and Tetrahedrons as D increases. In fact, the sequence is defined, not by a function but by a procedure: the Arithmetic Triangle.

1	0	0	0	0	0	0
1	1	0	0	0	0	0
1	2	1	0	0	0	0
1	3	3	1	0	0	0
1	4	6	4	1	0	0
1	5	10	10	5	1	0
1	6	15	20	15	6	1

This, however leads to another question: what is the Geometric interpretation of the column of 1’s at t= he left of the triangle?

In order to answer this question we must look to another aspect of the Arithme= tic Triangle: it’s use in describing the ways = in which a Set may be partitioned.

Arithmetic Triangle Showing Ways to Choose "P" Elements from a Set Containing "N" Elements


	P=3D0	P=3D1	P=3D2	P=3D3	P=3D4	P=3D5	P=3D6	P=3D7
N=3D0	1	0	0	0	0	0	0	0
N=3D1	1	1	0	0	0	0	0	0
N=3D2	1	2	1	0	0	0	0	0
N=3D3	1	3	3	1	0	0	0	0
N=3D4	1	4	6	4	1	0	0	0
N=3D5	1	5	10	10	5	1	0	0
N=3D6	1	6	15	20	15	6	1	0
N=3D7	1	7	21	35	35	21	7	1

“The Empty Set is a proper subset of all sets.”

In the above chart, N=3D0 represents the empty set. Since it is a proper subset of all= sets, its presence in the Arithmetic Triangle indicates that there is 1 way of choosing zero elements from any set. But, this factor is not included i= n the chart we have developed for calculating the characteristics of the set of L= IGU in N dimensions. In order pro= perly to include the column of 1’s (which is the necessary initiator of the Arithmetic Triangle) we must assume a D of less than 0!

The Geometric analog of the empty set can only be the Void. This is not an Esoteric statement, although the necessity of the existence of the Void was deduced = by the Ancients. As Lucretius wr= ote in “De Rerum Natura”;

“But yet creation&#= 8217;s neither crammed nor blocked

About by body: there̵= 7;s in things a void – …

…For were it not, t= hings could in nowise move;…”

Oriental Wisdom tells us;

“Look, It cannot be seen – It is beyond form.

Listen, It cannot be heard – It is beyond sound.

Grasp, It cannot be held – It is intangible;

Therefore they are joined= in One.

…

Knowing the Ancient Begin= ning is the Essence of Tao.”

= T= ao Te Ching (14)

<= /p>

Modern Physics tells us t= hat if the nucleus of an atom were the size of a soccer ball, there would be Void = for about 5 miles in all directions before one encountered the nearest electron cloud. Therefore, we propose = adding to the list of Dimensions one preceding D=3D0; D=3DNull which has for its content the Void. Since the Void underlies all manifestations of D, it is properly included in the list of the components = of each manifestation of D, and gives us the complete Arithmetic Triangle as t= he generator of our listing of the structure of the Set of LIGU in N Dimension= s.

<= /p>

<= /p>

Arithmetic Triangle Showing Structure of Least Inclusive Geometric Unit in N Dimensions<= /o:p>

Void

Points

Line Seg

Triangle<= /span>

Tetrahedron

Hyper-Tet

????????<= /span>

????????<= /span>

????????<= /span>

D=3DNull

1

0

0

0

0

0

0

0

0

D=3D0

1

1

0

0

0

0

0

0

0

D=3D1

1

2

1

0

0

0

0

0

0

D=3D2

1

3

3

1

0

0

0

0

0

D=3D3

1

4

6

4

1

0

0

0

0

D=3D4

1

5

10

10

5

1

0

0

0

D=3D5

1

6

15

20

15

6

1

0

0

D=3D6

1

7

21

35

35

21

7

1

0

D=3D7

1

8

28

56

70

56

28

8

1

Now we shall review the structure of the Set of LIGU in N Dimensions.

<= /p>

For D=3DNull, we have the Void. = [ = ]

<= /p>

For D=3D0, we have the Po= int. &= nbsp; &nbs= p; .= = &nb= sp;

<= /p>

For D=3D1, we have the Li= ne Segment.

&n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp;

&= nbsp;

For D=3D3, we have the Tetrahedron. = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; =

&= nbsp; &nbs= p; &= nbsp;

= &nb= sp; = &nb= sp; =

= ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; = &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; =

There is an interesting parallel between the structure of the LIGU for D=3D4 and = the Esoteric Structure of the Human Being:&nbs= p; The Hyper-Tetrahedron consists of 5 nested solids defined by 5 point= s, while the Esoteric Structure of the Human Being consists of 5 bodies nested together; Physical, Energic, Astral, Mental and= Spirit.

It is of further interest to note that the characteristics of the Set of LIGU = in N Dimensions can easily be represented in 2 dimensions by the set of regular geometric polyhedrons with lines interconnecting their vertices.=

D=3DNull = [ = ]

D=3D0 = &nb= sp; = .=

D=3D1 = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; = D=3D2 &= nbsp; &nbs= p; &= nbsp;

D=3D3 = &nb= sp; =

D=3D4 &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp;

<= /p>

= ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; &= nbsp; &nbs= p; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; &= nbsp; &nbs= p; &= nbsp; &nbs= p;

D=3D5 = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = ; &n= bsp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; = &nb= sp; &= nbsp; &nbs= p; &= nbsp;

In Esoteric terms, these figures may be interpreted to indicate that Mankind is indeed a fourth spatial dimensional being, composed of five bodies intermingled, and that the pentagon enclosing the pentagram models the interconnections of these bodies. This may account for the power of = those geometric figures in “Magic” ritual and preparations.

Furthermore, the figure for D=3D5 can be interpreted to show that 6 “Worlds”= exist within the encompassing scope of a higher Entity. Note that this diagram contains the symbol for the Principle of Correspondence as well as the Unity of Opposites within the encompassing Hexagon. Also, it can model the relationships of the lower 6 Chakras within t= he 7^th.

Apology

When I was researching historical documents in preparation for the writing of th= is report, it came to my attention that the ancient Greeks were aware that the Arithmetic Triangle describes the structure of the Tetrahedron; and subsequ= ent thinkers have noted that the Triangle describes the structure of the Hyper-Tetrahedron (although I have been unable to find the exact attribution for this discovery.)

Therefore, I must confess that this effort at best consists of an independent rediscov= ery of things previously understood. However, I can take comfort that this places me in Noble company, since Pascal himself unwittingly rediscovered applications of the Arithmetic Triangle well known to the 5^th Century Chinese and later the Hindu and Persian mathematicians.<= /span>

No doubt the conjectures I have made regarding Esoteric implications of this geometry are well known and in greater depth by Advanced Initiates. In fact, with hindsight, and as of= this writing, the only original contribution I can lay claim to is the definitio= n of the Set of Least Inclusive Geometric Units in N Dimensions. Even that, I acknowledge is subjec= t to review.

Arithmetic Triangle Showing Structure of Least Inclusive Geometric Unit in N Dimensions<= /o:p>


	Void	Points	Line Seg	Triangle<= /span>	Tetrahedron	Hyper-Tet	????????<= /span>	????????<= /span>	????????<= /span>
D=3DNull	1	0	0	0	0	0	0	0	0
D=3D0	1	1	0	0	0	0	0	0	0
D=3D1	1	2	1	0	0	0	0	0	0
D=3D2	1	3	3	1	0	0	0	0	0
D=3D3	1	4	6	4	1	0	0	0	0
D=3D4	1	5	10	10	5	1	0	0	0
D=3D5	1	6	15	20	15	6	1	0	0
D=3D6	1	7	21	35	35	21	7	1	0
D=3D7	1	8	28	56	70	56	28	8	1