We offer surprises about     

in our e-book       Prime Passages to Paradise

by H. PeterAleff





Footnotes :



1 Quote attributed to

Don Zagier in Dominic Olivastro: "Ancient Puzzles", Bantam Books, New York, 1993, page 15 middle






2 Sybil P. Parker and

Jay M. Pasachoff: "McGraw-Hill Encyclopedia of Astronomy", Second Edition, New York, 1993, page 139 left.







  Calendar - Clock for Earth


embedded in the number world    


You are on page

Prime calendar clock

  0  1   2   3   4   5   6   7   8     + 10   11  12  13  14  15  16  17  18

Our planet may have lost its ancient position at the center of the physical world, and it may now appear to be just one of many specks of dust in just one of many galactic swirls.  Yet, it remains distinguished among probably all others in this universe by having its unique orbital signature enshrined as a prime feature at the center of an important facet in the number world.

Primes are numbers that cannot be divided evenly, except by themselves and by one, and they are therefore the basic building blocks of the number world from which all other numbers are composed.

Some mathematicians with a mystical leaning have called primes "the inexplicable secrets of creation"1.  Others held and hold that the laws behind their distribution are one of the greatest mysteries and possibly beyond human understanding.

On the other hand, as we saw in the discussion of number pyramids, prime numbers line up in striking patterns when we write the sequence of natural numbers not as a continuous line but in layers that go, for instance, from one square to the next and so form the silhouette of a pyramid. 

Those alignments of primes suggest that this way of writing the numbers reveals them in their "natural habitat", along the inherent "grain" of the number world.  The square-to-square pyramid contains the longest alignments in all the arrays of its type, so it seems to show the preferred direction of that grain.  And this preferred way of presenting the numbers happens to give us the orbits of our earth and its moon, with two unusually prime-rich circles that are positioned prominently at the center of its apex.

The first of these circles is centered directly on the vertical under zero, next to the pyramid's mid-line under the number one.  You can see it on the poster of the square-to-square pyramid and on the detail view below that.

Squapyr2.gif (90876 bytes)

Poster of the square- to- square number pyramid with the upper prime-rich circle.   Gif = 89 Kb

view vector version  533 Kb (= long download)  for sharper details with the free vector-viewer plug-in from

Squaclock2optim.gif (70619 bytes)

Figure 15 :  Detail from the center of the pyramid apex in the poster, with the upper prime- rich circle that hits 13 primes in the 22 odd spaces it traverses.   Gif = 69 Kb
View vector version  52 Kb

That upper circle is centered in the upper half of the 930 box and has a radius of 5Ĺ units. It groups 13 primes in the 22 odd- numbered spaces it traverses -- that is 59%, or almost twice the 33.6% overall density of primes among the odd numbers below one thousand.

Clockfa1.gif (34076 bytes)

Figure 16 :  The second prime- rich circle of the "number clock", also centered in the square-to-square number-pyramid array. Gif = 34 Kb
view vector version   69 Kb

The second circle, shown in Figure 16, is similarly prime-dense.  It surrounds the middle of the upper left quadrant in the prime space 6,971, immediately left of the zero line. That larger ring has a radius of 12 and goes through or touches 25 primes in the 52 odd- numbered boxes it traverses.  That is just over 48% primes

The surroundings of that lower circle contain less than the overall 24.58% primes among the odd numbers up to 10,000, so this circle has again about double the prime-density of its surroundings. I continued my manual checking to about 14,000 but found no further prime-rich circles in that small apex, or in the other arrays described in the e-book. On the other hand, no one knows what other curious features may still be hidden farther down.

These two prime-rich rings together resemble a pendulum clock affixed to the central pillar in that number-pyramid apex, though with the small pendulum weight on top and the usually larger lock face below. The clock hands on that face are two prime-formed radii in the lower and larger circle that go from a triangular number near its middle to another triangular number near each oneís end.  They are frozen at ten minutes to eight and will therefore never show our changing time of day.

However, the circle centers contain the digit sequences for the reasonably constant lengths of our current months and years, as measured against the stars by our planetís daily rotation. 

Following the "grain" in this array that goes from square to square, we take the square roots of the two circle centers and divide the smaller into the larger. Ten times that ratio is 27.378 and so approximates the number of days in our sidereal month. That is the average time the moon takes to return to the same background of stars.

This "star month" is currently about 27.322 days long, so the number calendar is 0.2% off.  This approximation is fair when you consider the notorious fickleness of our lunatic satellite and the lack of flexibility among the numbers..

Now invert that ratio between the circle center roots and behold its even closer agreement with our own orbit around the sun. A thousand times that reciprocal works out to the number of mean solar days in our sidereal year, with an annual difference of less than five minutes:

10  x

6971  /  930
Days in sidereal

= 27.378274
= 27.321661


0.207 %

1000  x

930  /  6971
Days in sidereal

= 365.25312
= 365.25636


0.000887 %

This "star" year is slightly longer than our "tropical" or "calendar" year of currently about 365.2422 days.  It marks "the true period for the revolution of the Earth around the sun" because it "is determined by the time interval between successive returns of the Sun to the direction of the same star"2.

So, even though our earth may no longer occupy the center of the physical universe, as it had long done for many ancients, you find its present orbital signature indelibly etched into a highly privileged position of the number world, near the center of its most prime-aligning "natural" facet, and close to its apex.

Those orbital ratios are said to result from a billard- like chance collision between Earth and a Mars-size object that knocked loose the matter which then became the moon. It seems therefore unlikely that other planets might have this same combination of satellite and rotation time and orbit ratios. Until such an exactly Earth-like planet turns up, we can thus safely say that our home planet is the uniquely distinguished number calendar capital of the universe.

And that puts us squarely back at the center, at least for now, and for the next few hundred thousand years before the orbital ratios drift away from their present precision in matching the universal number calendar-clock.


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