# 70 x 7 AND 666 IN PASCAL’S TRIANGLE

I’ve discovered something that may account for the biblical “seventy times seven” and also, Revelation’snumber of the beast” which is 666.

SEVENTY TIMES SEVEN (70 x 7)

Repeating triangles within triangles reveal factors of 70 x 7.

In Pascal’s Triangle, it would seem that a cascade of 7-factors starts with each seventh line (and there is a similar phenomenon for each of the primes??) The numbers with 7-factors form an inverted triangle within the Pascal’s Triangle, six figures across and six down. At each additional line which is a multiple of seven, the number of such inverted triangles increases by 1??  Within the first inverted triangle of 7-factors (blue triangle in the drawing above), is another triangle whose corners sum to 490 or “70 times 7” (red). Thus, on the seventh line (which sums to 2 to the 7th power or 128), there are numbers containing factors of 7, which continue downward and form an inverted triangle comprising 21 values (7 x 3), and within which triangle is another triangle whose corners sum to 70 x 7 (210 + 210 + 70 = 490 = 70 x 7).

On scratch paper, I find that this pattern may continue. In the two inverted triangles of 7-factors which start at line 14, I calculate as I did above, that is, I add the center value on line 2 of an inverted triangle to the end values on its line 4 (the corners of the triangle-within), and I get a factor of 7-cubed!!! (1365 + 12376 + 2380 = 16121 = 7 x 7 x 7 x 47). Am I tempted to continue the Pascal’s Triangle to see what happens at line 21? Well, OK.

What happens in this third iteration is that there are three inverted triangles of 7-factors as expected, with the same dimensions as above. In the center inverted triangle, the summed corners of the triangle-within produce a factor of 70 x 7!!! The two side triangles-within produce factors of 7-squared.

Here are the calculations:

Two side triangles-within: (7315 + 134596 + 10626 = 152537 = 7 x 7 x 11 x 283)

Center triangle-within: (705432 + 2496144 + 2496144 = 5697720 = 70 x 7 x 2 x 2 x 3 x 3 x 17 x 19)

I still don’t know why the biblical authors would chose to cover up the factors of 70 x 7, allowing them to be released only with application of Sower’s numbers (100, 60, 30).

MARVELS OF PASCAL’S TRIANGLE

As you may know, Pascal’s Triangle starts with a tip top 1 (drawing further below), and each number below is the sum of the two directly above. Each of the rows sum to a power of 2. The lines in my drawings are numbered to show which power of 2. For example, line 7 sums to 2 to the 7th power or 128.

If you start at the top of a Pascal’s Triangle and trace diagonals carefully, you will find they add up to the values in the Fibonacci series (1, 1, 2, 3, 5, 8, 13, 21, 34, 55 . . . .).   In that series each value is the sum of the previous two values. That’s marvelous!

I had been looking for something in nature or in math that would be marvelous enough to be the foundation for the biblical 70 x 7’s (Matthew 18:22 (footnote NRSV) and Genesis 4:24). Perhaps I’ve found it. I guess the idea of a triangle that produces powers of two, and has within it a triangle within a triangle that produces powers of sevens or 70 x 7’s, might appeal to ancient peoples or to anyone for that matter. I have to wonder if the two interlocking triangles (one upright and the other inverted) on today’s flag of the State of Iσrαεl have anything to do with this.

THE NUMBERS OF THE BEAST (666 AND 616)

I know I am in the right neighborhood with the Pascal’s Triangle because some numbers from Revelation can be summed up there in a nice symmetrical way. For instance, Revelation’s number of the beast, that is, “666,” is the sum of corners of three triangles, each sized like the triangle above that yields the 70 x 7: ((210+28+45) + (210+28+45) + (70+15+15) = 666). The three triangles (gold in the drawing below) are hinged on the center 70 x 7 triangle-within (red). A chevron (pink) crisscrosses the center triangle (red) and sums to the alternate number of the beast, 616 which is in some manuscripts (56 + 56 + 126 + 126 + 252 = 616).

Additionally, sums of other Revelation numbers make for nice symmetrical patterns in the Pascal’s Triangle:

1260 = 924 + 252 + 70 + 7 + 7

1600 = 792 + 792 + 8 + 8

Perhaps the Divine spent a million eons making the Pascal’s Triangle and then decided to do something totally unexpected and totally unimaginable and totally unpredictable and began making me and you.

HEXAGONALS YIELD 12-FACTORS

I am beginning to suspect that the Pascal’s Triangle does something marvelous with 12-factors also, and that may be why 12 is a favored number in the Bible. In the next drawing, the numbers which ring the center number “70” form hexagons or if you like, hexagram stars. Does this hexagonal shape remind me of anything, such as a certain flag with a six-pointed star?

Here are the calculations for the hexagonals resulting in 12-factor sums:

Yellow star:

70 + 2(35) + 2(56) + 2(126) = 504 = 12 x 7 x 6

Blue star:

70 + 20 + 252 + 2(21) + 2(84) = 552 = 12 x 2 x 23

Deep yellow star:

70 + 2(15) + 2(28) + 2(210) = 576 = 12 x 12 x 4

Pink star:

70 + 2(5) + 2(8) + 2(330) = 756 = 12 x 7 x 3 x 3

Green star:

70 + 924 + 2(45) + 2(1) + 6 = 1092 = 12 x 7 x 13

CAN MATH EVER BE FUN?

Isn’t it a shame that in schools these days, math is not allowed to be fun, and anything that could be sacred math is just swept away and replaced by bits and pieces of math that is supposedly “useful” to industry but which hardly anyone uses. The only thing children learn from that is to hate math.

WHO DISCOVERED PASCAL’S TRIANGLE?

Was Pascal the first to discover this triangle? Maybe he honestly thought he was and his contemporaries believed him. It seems rather likely that the biblical authors got there before him.

Updated January 2, 2015 to add the following:

EVER MORE HEXAGONALS IN PASCAL’S TRIANGLE

If I sum all the numbers that comprise the five six-pointed stars above (counting the “70” only once), I get 3200, which is 2 to the seventh power times 5-squared. Is seven of twos (2 x 2 x 2 x 2 x 2 x 2 x 2) as sacred as two of sevens (7 x 7)?

I find that each of the 21 values in the inverted triangle of 7-factors (first iteration) can serve as the center of a hexagonal ring where the sum of seven numbers is evenly divisible by 12 or 144 (12 x 12). However, based on my survey, it appears that none of the non-seven-factor numbers in the near vicinity can do this; at least one exception being the topmost “2.” Not all of the 21 values can serve as the center for larger and larger hexagonals as can the “70,” as not all can even host a second hexagonal. So far, only the “70” and each “210,” the corners of the triangle-within that sums to “490,” have a factor of 144 (12 x 12) for a hexagonal plus center sum.

I might conclude from this that the twelves, sevens, and seventy times seven’s that had special meaning to the authors of the Bible, were derived from the Pascal’s Triangle.

Online you can find a generator of Pascal’s Triangle rows at:

http://mathforum.org/dr.cgi/pascal.cgi?rows=58

I generated the first 58 rows and started looking for factors of seven, 70 x 7, 7-cubed, twelve, and twelve-squared in the seven inverted triangles starting at row 49 (7 x 7) in hexagonal and in triangle-within patterns.

I went into my kitchen and there was a tray I’ve had forever and I looked at it and it has seven concentric hexagrams plus some decorations around the edge in 12-count and 24-count. So where are the seven concentric hexagrams in the Pascal’s Triangle? I don’t know, but I am glad I found as many as I did.

Updated January 4, 2015 to add the following:

I found a set of six concentric hexagonals centered on the center value of row 26 (10400600), a mix of upright hexagonals and those rotated 90 degrees, just as above with the set of five hexagonals.

Summary of sum-of-seven results (hexagonal plus center):

Center value = 10400600

Ring 1 (rotated); sum = 10400600 + 2(20058300) + 2(9657700) + 2(5200300) = 80233200 = factor 144

Ring 2 (rotated); sum = 10400600 + 2(37442160) + 2(7726160) + 2(2496144) = 105729528 = factor of 12

Ring 3 (rotated); sum = 10400600 + 2(67863915) + 2(5311735) + 2(1144066) = 159040032 = factor of 12

Ring 4 (point up); sum = 10400600 + 2704156 + 40116600 + 2(4457400) + 2(17383860) = 96903876 = factor of 12

Ring 5 (point up); sum = 10400600 + 705432 + 155117520 + 2(1307504) + 2(21474180) = 211786920 = factor of 12

Ring 6 (point up); sum = 10400600 + 184756 + 601080390 + 2(245157) + 2(20030010) = 652216080 = factor of 12

Can a set of seven concentric hexagonals be far away?

PATTERNS OF 7-FACTORS WITHIN PASCAL’S TRIANGLE

Another interesting online resource is a calculator that generates the patterns of factors in the Pascal’s Triangle. Go to http://oldweb.cecm.sfu.ca/organics/papers/granville/support/pascalform.html and set “modulus” to 7 and rows to the maximum 100, click submit.

Above I have made my own drawing adapted from that website. There are 100 rows, that is, through row 99. How to count the rows: the first row is named “zero” because the rows of the Pascal’s Triangle sum to powers of 2 and are named for the powers. For example, the 8th row is row 7, because it sums to 128, 2 to the 7th power.

I could not have foreseen the perfection in this design which is that at row 49 (7 x 7) every value in that row (other than the two ends) is apparently evenly divisible by 7 and starts an inverted triangle, 48 values across and 48 rows down. It appears that these larger inverted triangles repeat at least into the next iteration where there are two side by side. Each of the neighboring larger upright triangles in the drawing comprise 21 (7 x 3) inverted triangles of 21 seven-factors each, and 28 (7 x 4) upright triangles of 28 non-seven-factors each, totaling 49 (7 x 7) triangles. Each larger upright triangle has 1225 values (7-squared times 5-squared), 49 values across and 49 rows up. The larger inverted triangle comprises 1176 values (12 x 7 x 7 x 2). No wonder sevens are special in the Bible!!! Other factors make pretty patterns but seven is the best I’ve found using this calculator.

I think it is fairly obvious that the biblical authors took many of their numbers from what would later be named the Pascal’s Triangle.

Updated January 14, 2015 to add the following:

AMAZING ZIGZAG SUMS PRODUCE FACTORS OF 12 x 12

What was the biblical authors’ fascination with 12’s and with 144 (12 x 12)? There is more to twelves than just the hexagonals in the Pascal’s Triangle explained above.

If you make a diagonal zigzag sum of the values in the Pascal’s Triangle, that is, add two values diagonally up, then add two more values diagonally down, continuing across the Triangle, the sum will have two factors, 144 and a power of 2. This seems to work indefinitely?? except for the initial sums at the top of the Triangle, which are still closely related to 12. The powers-of-2-factors increase by 2 with every zigzag sum. There are two progressions, depending on whether the center value of the row is the peak (highest) or nadir (lowest) point of the zigzag. Incredibly, the sum of the zigzag where the center value is at a peak, is equal to the sum of the zigzag of the following center value at a nadir. The first center value where you get a result of a 144-factor for both peak and nadir is “70.”

Here are the sums and factors of center values, 70 through 705432:

70 peak 1152, 2^3 x 144

70 nadir 288, 2 x 144

252 peak 4608, 2^5 x 144

252 nadir 1152, 2^3 x 144

924 peak 18432, 2^7 x 144

924 nadir 4608, 2^5 x 144

3432 peak 73728, 2^9 x 144

3432 nadir 18432, 2^7 x 144

12870 peak 294912, 2^11 x 144

12870 nadir 73728, 2^9 x 144

48620 peak 1179648, 2^13 x 144

48620 nadir 294912, 2^11 x 144

184756 peak 4718592, 2^15 x 144

184756 nadir 1179648, 2^13 x 144

705432 peak 18874368, 2^17 x 144

705432 nadir 4718592, 2^15 x 144

Is there an intelligent Designer? If it takes intelligence to recognize these number patterns, then does it take intelligence to create them? Or should we argue that these patterns were not created; rather, they just happen to be and nobody of any intelligence should even bother to look at them. Maybe there is no intelligent life on Earth and we are all just something that happened to be.

If we found such number patterns being beamed at us in energy waves of some sort from the far reaches of the Milky Way, wouldn’t there be a lot of excitement that at last we had found intelligent life outside our Solar System? Oh, maybe not. Many people seem to be so – um – lethargic, like they are sleep-walking through life. People, just go back to staring at the TV and clicking on your Facebook. We’ll give you a sound bite if anything important ever happens that we need you to know about.

I doubt I’m the first to notice the zigzag sums, given the emphasis in the Bible on 12’s and 144 (12 x 12). Why didn’t someone teach me about this?

FIBONACCI NUMBERS SUMMED IN PASCAL’S TRIANGLE

This drawing shows Fibonacci numbers summed in diagonals in Pascal’s Triangle.

Updated January 31, 2015 to add the following:

THE 30 60 100 CONNECTION

It appears that the ancients multiplied with the Sower’s parables numbers (30 60 100 or 100 60 30) in order to reveal a factor of 70 x 7 (special numbers in the Bible). It also seems that they set up number sets intentionally so that a special “70 x 7” would be revealed. At least it is starting to look like it was intentional, now that I have published 11 examples of a 70 x 7 factor (see sidebar for “Sower’s Sevens” and “Sevens” and “100 60 30”).

Just to be sure I wasn’t misjudging the rarity of 70 x 7’s, I took random numbers from the phone book and applied 30 60 100 (Sower’s numbers) to check for the occurrence of 70-factor.  (The Sower’s add a factor of ten.)  The expected result is that a 70-factor will happen 1 in 7 or 7 in 49.  The phone book yielded 4 in 49, much less.  So likely I did not have a representative sample size.  A 70 x 7 is expected 1 in 49.  I found NONE.  But it is reassuring that the Sower’s sevens (70 x 7) seem to be at least as rare as I had theorized, even though not rare in ancient books.  I seem to find them almost every time I try, but then again, I haven’t been keeping count of the negative results, not that the sevens would have to be there every time.

I believe the Sower’s numbers (30 60 100) have something to do with the so-called Pascal’s Triangle. There multiplication with Sower’s parables numbers generates a much greater rate of 70 x 7’s, often in clusters. The rate of unexpected 7-results among 51 sets, from rows 0 thru 13, three values in a row or three values in a triangle, with application of 30 60 100, is approximately 37 percent, much higher than the expected 1 in 7, with 70’s,  six of 70 x 7 and two of seven-cubed.  (This count does not include any number sets composed of all 7-factor numbers, except where there are unexpected 7-results.)  So it does seem probable that the Sower’s numbers refer to a phenomenon of emerging 70 x 7’s in the Pascal’s Triangle which they reveal.

Example – how to use the Sower’s parables numbers

Here is an example of how to use the Sower’s parables numbers to reveal 70 x 7 in the Pascal’s Triangle:

Three values in a triangle from rows 10 and 11 are: 1, 1, 11

Multiply by 100, 60, 30:

1 x 100 = 100

1 x 60 = 60

11 x 30 = 330

Sum of products = 490 = 70 x 7

Only a larger sample size will tell me with reasonable certainty how comparatively abundant the 70 x 7 factor is when revealed by manipulations with 30-60-100 in the Pascal’s Triangle. But I’m going to put this up anyway as I don’t plan to be doing the thousands and thousands of calculations that would be necessary. Maybe a computer programmer could set up a program to discover more.

By the way, I also did Sower’s calculations for 21 sets of 3 prime numbers each (excluding 1 through 7) to see if I was missing something about what drives the production of sevens in sacred texts (other than someone embedding the sevens deliberately). Nothing unusual – two results had a factor of 70.

70^7 AND MORE FROM THE TALPIOT SYMBOL – update April 9, 2015

Here I discover two big numbers from the Book of Revelation, 1,260 and 1,600, so maybe I am in the right neighborhood. Also, a new way of looking at 70 x 7. Does it point to 70 to the 7th power?

Multiply the numbers that form the outline of the Talpiot symbol around a center 70 in the so-called Pascal’s Triangle. For those of you who don’t know, the Talpiot symbol is the circle and chevron found on what might be the tomb of Jesus at Talpiot.

Centered on 70:

70 x 126 x 126 x 56 x 56 x 35 x 35 x 36 x 36 x 28 x 28 x 21 x 21 x 15 x 15 x 10 x 10 x 6 =

258,252,595,599,768,698,880,000,000

This Talpiot symbol number, interestingly enough, is evenly divisible by 1,260 to the sixth power or 4,001,504,141,376,000,000. The value 1,260 is mentioned twice in the Book of Revelation (11:3 and 12:6).

The Talpiot symbol number is also evenly divisible by 1,600-cubed.   The value 1,600 is mentioned in the Book of Revelation (14:20).

The Talpiot symbol number is also evenly divisible by 12 to the tenth power or 61,917,364,224, twelve being a special number in the Bible.

The Talpiot symbol number is also evenly divisible by 70 to the seventh power (8,235,430,000,000). Does this sound anything like a biblical “seventy times seven” to you? More like 70 x 70 x 70 x 70 x 70 x 70 x 70, that is, seventy multiplied by itself seven times.

Seven is a favored number in the Bible. The biblical “70 x 7” is found printed in Matthew 18:22 (footnote NRSV) and also in Genesis 4:24.  By the way, I believe that means seventy times sevenfold (DRA), not 77 times. And now, after my experience here, I believe it may point to 70^7, that is, 70 to the seventh power.

490 (or 70 x 7) to the fifth power is also a factor.

Centered on 153:

If you do a Talpiot-style calculation with the 153 mentioned in John 21:11, you get a number evenly divisible by 153 to the seventh power. The value 153 is found on row 18 of the so-called Pascal’s Triangle (remember the first row is row 0).

153 x 969 x 171 x 18 x 816 x 136 x 17 x 1 x 1 x 1 x 1 x 1 x 1 x 15 x 120 x 680 x 3060 x 11628 =

37,494,682,400,900,604,659,466,240,000, evenly divisable as noted and also, by 12^9.

Centered on 20 (flipped):

What happens if I do an upside-down Talpiot-style calculation centered on Pascal’s 20?

20*10*15*35*1*6*21*56*126*10*15*35*1*6*21*56*126*252 = 109,801,273,639,357,440,000,000 is the product and it is evenly divisible by 70 to the seventh power.

490 (or 70 x 7) to the fourth power is also a factor.

Centered on 70 (flipped):

An upside-down Talpiot-style calculation centered on Pascal’s 70 follows: 70*35*56*126*7*28*84*210*462*35*56*126*7*28*84*210*462*924 = 10,065,063,046,289,956,884,525,691,699,200,000 and is evenly divisible by 70^5, 1260^5, 1600^2, 616^3, 12^11, 490^5. Calculations for this paragraph are from the 50-place calculator at http://keisan.casio.com/calculator.

Centered on 666:

Here is a Talpiot-style calculation done with 666 in row 37 and the calculator mentioned in the previous paragraph.  666 is Revelation’s number of the beast (13:18).

666*37*7770*36*630*703*8436*501942*66045*7140*595*34*1*1*1*1*1*1 =

123316018492769915210428153495276800000

This number is evenly divisible by every number 1 through 10. Also, evenly divisible by:

666^5, 70^5, 1260^5, 1600, 153^5, 12^5, 490^3.

So yes, it looks like this is the right neighborhood for these biblical numbers.

The Talpiot-style calculation involves the multiplication of the product of seven numbers in the circle times the product of eleven numbers in the chevron. Do the numbers 7-11 remind you of anything?

But how could the ancients have done such calculations without an electronic calculator?

Update on May 9, 2015

THE TRIANGLE’S POWERS OF TWO PRODUCE SOWER’S SEVENS

This may be what I’ve been looking for these many years!!! Although it doesn’t overturn what I said above, it is far more exciting.

It shows perhaps, the origin of the biblical “seventy times seven” and the Sower’s numbers in the Gospel (100-60-30 and 30-60-100).

Just a reminder, the rows of the Pascal’s Triangle sum to increasing powers of 2.

These powers of 2 produce Sower’s Sevens. Here is how.

The leftmost column in the chart below shows, obviously, the powers of 2 in ascending order.

Each number in the second column is the sum of the number above and the number to the left.

The third column shows that each third value in the second column is evenly divisible by 7. It would seem that this pattern may continue with each seventh result of a seven-factor being instead 7-squared.

The fourth and fifth columns in the chart are the sum of products obtained by multiplying three consecutive values in the second column with Sower’s numbers (100-60-30 and then 30-60-100). For example, (15 x 100) + (31 x 60) + (63 x 30) = 5250 (factor of 70); and then (15 x 30) + (31 x 60) + (63 x 100) = 8610 (factor of 70). If the three consecutive values in the second column are selected so that the third value is the value evenly divisible by 7, then the sum of products will be evenly divisible by 70. It would seem that every seventh sum of products yields a biblical factor of seventy times seven.

The fifth column divided by the fourth column approaches an approximate golden ratio if you don’t mind rounding up to 1.618 (for example, quotient in bottom row is 1.6176471411150741660343138936826).

0

1                              1

2                              3

4                              7                              /7=1                       490 (70 x 7)         910 (70)

8                              15

16                           31

32                           63                           /7=9                       5250 (70)              8610 (70)

64                           127

128                         255

256                         511                         /7=73                    43330 (70)           70210 (70)

512                         1023

1024                       2047

2048                       4095                       /7=585                  347970 (70)         563010 (70 x 7)

4096                       8191

8192                       16383

16384                    32767                    /7=4681                2785090 (70)       4505410 (70)

32768                    65535

65536                    131071

131072                  262143                  /7=37449             22282050 (70)    36044610 (70)

262144                  524287

524288                  1048575

1048576                2097151                /7 x 7 =42799      178257730 (70)  288358210 (70)

2097152                4194303

4194304                8388607

8388608                16777215             /7=2396745         1426063170 (70 x 7)         2306867010 (70)

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