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Geometric Properties
As before, the world was created with wisdom - and not just any wisdom, but perfect, divine wisdom. No surprise then that the opening verse also contains tremendous wisdom in its geometric properties and balances.Here is a brief introduction to this vast subject.
Some numbers are called "figurate numbers". They can be shown geometrically by taking points, or dots, and arranging them into a regular shape such as a triangle or a pentagon.
As we saw, Genesis 1:1 consists of 28 letters and the first word, Beresheit, has 6 letters. 6 and 28 are the 3rd and 7th triangle numbers respectively. These numbers 3 and 7 are at the core of Genesis 1:1
| Triangle Numbers | |||
| T1 | 1 | = | 1 |
| T2 | 1+2 | = | 3 |
| T3 | 1+2+3 | = | 6 |
| T4 | 1+2+3+4 | = | 10 |
| T5 | 1+2+3+4+5 | = | 15 |
| T6 | 1+2+3+4+5+6 | = | 21 |
| T7 | 1+2+3+4+5+6+7 | = | 28 |
| T8 | 1+2+3+4+5+6+7+8 | = | 36 |
| T9 | 1+2+3+4+5+6+7+8+9 | = | 45 |
| T10 | 1+2+3+4+5+6+7+8+9+10 | = | 55 |
| .. | ... | .. | |
| T37 | 1+2+3..+37 | 703 | |
| .. | ... | .. | |
| T73 | 1+2+3..+73 | 2701 | |
(Triangle numbers are called by R.Moshe Kordevero "source numbers" (Mispar Mekori), i.e. a number formed from a beginning. Notice above how the number 1 is always at the head of a triangle number. Thus they hint that everything stems from the One, i.e. G-d [1])
Triangular numbers can be arranged in Triangle formation such as:
Genesis 1:1 has Gematria of 2701
2701 is the 73rd triangular number:
Genesis 1:1 has 28 letters. 28 is a triangular number.
The base of this triangle - ואת הארץ has Gematria of 703
703 is the 37th triangle:
| Center Point | ||
37 is the exact center point of 73
|
(i.e. for 73 dots side by side, the middle dot is the 37th)
Thus, T37 fits perfectly inside the center of T73:
Infinite Triangles
The letter expansion (Milui) of the base ואת הארץ = 703 gives 2701!| Letter Expansion of ואת הארץ | ||||||
| ץ | ר | א | ה | ת | א | ו |
| צדי דלת יוד | ריש יוד שין | אלף למד פי | הי יוד | תאו אלף ואו | אלף למד פי | ואו אלף ואו |
| 558 | 890 | 275 | 35 | 531 | 275 | 137 |
| Total=2701 | ||||||
Due to this symmetry, the same pattern can be repeated again.
And again and again like an infinite mirror.
All this infinite expansion from the symmetry in the verse!
Note that we could also have mirror-reflected the 703 triangle Milui (letter expansion) to make an infinite star of David pattern.
Interestingly, the 703 triangle is geometrically related to the 10th "star of David" number = 541 = "Yisrael".
For, both the 703 triangle and the 10th star of David have 37 rows and both share the same perimeter count of 108 dots (more on 108 another time, be'H).
Interestingly, the 2701st Triangle (1+2+3+..2701) = 3,649,051
| Sum of Thousands | 3+649+051 | = | 703 |
| Perimeter | 8,100=8+100 | = | 108 |
3-D Tetrahedrons
The outline of the inner 703 triangle also hints to a 3-D shape, namely, this 3-D tetrahedrons structure.The 4 Tetrahedrons (pyramids) of size 36 + the 703 base gives:
4*8436 + 703 = 34447 = 49 * 703 dots.
The Tetrahedrons hint to the hidden Merkaba of Shechinah (Divine presence), the Merkaba of Peace as we will discuss another time, be'H.
The 4 combined tetrahedrons also form a single tetrahedron, thus the pattern can be repeated in the same way infinitely.
Other Arrangements
There are many other interesting ways to arrange the 2701 dots of Gen.1:1, such as according to the balance of words.A few examples (discovered by Vernon Jenkins[2]):
Another example:
These former two are not so a-priori though. The following one is perhaps more interesting:
Same thing as previous but easier to see:
Triangle - the Prime Polygon
Note that the Hebrew word for Triangle is Meshulash - משולש.The triangle is the most prime polygon and thus has many connections to wisdom and Genesis 1:1.
The Gematrias of משולש are:
| Standard Gematria | = | 676 | Ordinal Gematria | = | 73 | Small Gematria | = | 19 |
Interestingly, 2701 also points to 676 as follows:
| Center Point | ||
1351 is the exact center point of 2701
|
| Center Point | ||
676 is the exact center point of 1351
|
676 is the final center point since only odd numbers have center points.
Thus, 2701 points to 676 and 676 = 262, where 26 is the Gematria of G-d's Name.
For the creation is all about the relationship between man and G-d
45 = אדם (man)
452 + 262 = 2701
Man is supposed to make G-d the innermost center point of his life and always be aware of Him (Shulchan Aruch 1:1). Then, man will infuse holiness in the entire creation and be a partner with G-d (Path of the Just ch.26, Shaarei Kedusha Gate 3).
Interestingly, the Ordinal Gematria of Genesis 1:1 (298) = The 26th Prime (101) + The 45th Prime (197)
Also interesting, we have seen previously that the Gematria of Torah = 611 and the first 611 digits of Pi sum to 2701. If we look at the 611th position of Pi:
| 4526356082778577134275778 |
| 611th position of Pi (notice 45 and 26) |
Wisdom Triangle
The world was created with "wisdom" (Targum Yerushalmi, Genesis 1:1). This is represented by the letter "Yud" which is like a point-dot.The Gematrias of "Wisdom" (חכמה) are:
| 73 | = | Wisdom | (Standard Gematria) |
| 37 | = | Wisdom | (Ordinal Gematria and center point of 73) |
| 19 | = | Wisdom | (Small Gematria and center point of 37) |
| 10 | = | Wisdom | (Small Gematria reduced, 1+9=10 and center point of 19) |
| 1 | = | Wisdom | (Small Gematria reduced reduced, 1+0=1 = the point of creation) |
Thus, the "wisdom" series is: 1,10,19,37,73
In triangular numbers:
| T1 | (1st Triangle) | = | 1 |
| T10 | (10th Triangle) | = | 55 |
| T19 | (19th Triangle) | = | 190 |
| T37 | (37th Triangle) | = | 703 |
| T73 | (73rd Triangle) | = | 2701 |
703 is the center triangle inside 2701 (see above)
190 is the center triangle inside 703
55 is the center point between 37 and 73. 55 appears in many other places too[3]
Star of David
Let us now look at some of the hexagram (star of David) properties of 2701.
Star of David figurates are made by combining two triangles. For example, 2 triangles of 28 to form a Star of David of size 37. Or 2 triangles of 55 to form a star of david of 73. (both 28/55 and 37/73 are important numbers in Genesis[4])Star of David numbers are given by the formula:
6n(n − 1) + 1
| Star of David (Hexagram) Numbers | |||
| ST1 | = | 1 | |
| ST2 | = | 13 | |
| ST3 | = | 37 | |
| ST4 | = | 73 | |
| ST5 | = | 121 | |
| ST6 | = | 181 | |
| ST7 | = | 253 | |
| ST8 | = | 337 | |
| ST9 | = | 433 | |
| ST10 | = | 541 | |
Since both 37 and 73 are star numbers and 2701=37*73, then it is possible to make a "star of star" figurate out of 2701 points.
This is the one and only "star of stars" whose stars follow each other and are center points of each other (73 follows 37 and 37 is the center point of 73) - a unique phenomena in all math!
Notice also the shape of the blank spaces between the stars. It is the same shape as the pattern we saw above in the word balances. shown here in red:
This page was meant to be a brief introduction. There are many, many more hints in the perimeters, midpoints, balances, etc. etc. in these figures and also other shapes such as the hexagons. The wisdom which springs from this verse appears to be inexhaustible.
All this reminds us of what is surely one of the most beautiful examples of design in the natural world - snowflakes! Until recently, their beautiful shapes were unseen as they can only be seen through a microscope. Notice how these actual snowflakes display a Star of David. This is very typical of snowflakes.
>> Next: 6 Days and Sabbath
Footnotes
- [1] from the book Otyot Machkimot by Rabbi David Leitner return to text
- [2] used with permission of Vernon Jenkins, picture by Bill Downie return to text
- [3] For example:
If the inside of a triangle is 55 (T10)
Then its perimeter is 13+12+11 = 36 = 6*6
And the perimeter of 2701 (T73) is 216 = 6*6*6
This is triangle 13 = 91 = Havaia Adni!
So the inside of the triangle of Echad/Ahava which is 91 havaia adni = 55!
190 is the center triangle inside 703
Just like 703 is the center triangle inside 2701 return to text - [4] 55 is the midpoint between 37 and 73 and 28 is the midpoint between 19 and 37. 19,37, and 73 are the Gematrias of "Wisdom" (Chachma) as we saw. 55 is also the 10th triangle number and 10th Fibonacci number. Interestingly, it is also the exact digit sum of the speed of light in meters, the standard international unit of length. This is probably very deep. return to text