Cube Unlike All Others
(I have refraining posting this because it does need to be edited)
R.L.G. wrote (from original post of Another Look at the Cipher of II 76)
Now let us incorporate the numbers of the Cipher into a pair of equations that reveal a very specific number for their result. We begin in the forward direction, and multiply all the numbers severally except the final one. Thus we have:
4 × 6 × 3 × 8 × 2 × 4 × 3 × 24 = 24⁴ = 331,776
I will leave it to John Griffith to expound the importance of the number 331,776, as it relates to a four-dimensional Hypercube. For the present purpose, it is only important to note that this product equals 24 to the fourth power.
Well obviously it has taken quite a while for me to get back to you on the hypercube. I am not sure whether the "cube in the circle" of verse 73 ought to be identified with the cube at the base of the "absolute dead center hypercube" of the "thinking now occuring" or not. Either way it is possible to identify a few relevant verses and passages with the mathematics of that cube. I have limited the account to bare essentials – but the post is still quite lengthy.
In verse 144 are written four words which equal the volume of a hypercube of edge length 6
name foursquare mystic wonderful = 1296 = 6⁴
The three dimensional base cube of the hypercube of volume 1296 is the 6 cube, and it is the only cube whose exterior surface area equals in value its volume
Volume of Cube of Edge 6 = 6³ = 216 = 6 × 6² = Surface Area of Faces of Cube of Edge 6
In regards to the relation of the absolute dead center cube or the cube unlike all others to the 6³ cube at the foundation of the 6⁴ hypercube, and which uniquely among the infinity of cubes is the only one whose surface area equals its volume, it is a topic which cannot be understood without a few preliminaries - to which I will limit myself today – going directly on from there to give an account of the relation of the absolute dead center cube to the cubit measure of the Great Pyramid – the Ahket or Horizon of Khufu – the monument designed by the architect Hemiunu according to the number of ipwt of the wnt of Djahuti recorded in the Revision archive at I͗wnw – Ἡλιούπολις – the City of the Sun. Hemiunu whose name signifies literally “Priest/Incarnation of I͗wnw, was a grandson of the pharoah immediately preceding to Khufu – Sneferu – and amongst his titles were "king's son of his body, chief justice, and vizier, greatest of the five of the House of Thoth"
Professor Leahy originally designed his cube to have a face diagonal length of 144 yielding the absolute dead center cube surface area
82944 = 8 × 10368 = 8 × ( 144 ÷ √2)²
There are eight square planes of area of the cube instead of just the six square planes of the faces because the absolute dead center cube is constructed with two interior planes orthogonally intersecting each other to measure the cube. The area of the unitary structure of the absolute dead center cube is the sum of the area of its surface and the area of its interior vertical and horizontal planes. The measuring action of these planes is analogous to the interior diameters of the circle squared hieroglyph intersecting at right angles measuring the circle through its orthogonal co-ordination of it.
In the philosophy of Leahy there is no nothingness, vanishing point, abyss – it is absolute embodied omnipotence perfectly responsive to the imperative of the beginning to create the world. The abyss of modernism is not unique to Crowley, it is an error into which descended much of the philosophy of subjectivity as initiated by Descartes and terminated by Nietzsche, Heidegger, and Derrida.
Notice if we apply the above principle of adding the area of the interior vertical and horizontal planes to the exterior surface area of the unique cube whose surface equals its volume the area of its eight planes equals √82944 = 1!2!3!4! the factorial of the factorials of one to four.
It is the sum of the columns, rows, and diagonals of the square of the Sun
One of the ways by which Doctor Leahy arrived at 82944 was through applying rational, linear, and integral product algorithms to series of words and/or numbers.
A rational product of a word, number, or sentence is the product of the ratio of its odd placed letters/numerals with its even placed letters/numerals.
A linear product is the product of its numerals or letters with each other
An integral product is the product of a linear and a rational product. It equals the square of the product of the odd placed numerals/letters of a word or number.
In applying these operations to decimal numbers Leahy discovered that there are only three numbers in the infinite series of natural numbers which equal their own integral products, viz., 1, 784, and 82944
integral(1) = (1)² = 1
i(784) = (7·4)² = 784
i(82944) = (8·9·4)² = 82944
On the basis of the unique existence of these integral products Leahy advanced the Theorem (dgleahy.com/p41.html)
"That the 4-dimensional hypercube volume whose edge is the volume/surface ratio of a base cube is to the base cube volume exactly as the boundary of the 4-dimensional hypercube of the base cube is to its volume — (x/6)⁴/x³ = 8x³/x⁴ — is uniquely true of the cube whose edge is √10368 (the “absolute dead center cube”)."
In the equation (x/6)⁴/x³ = 8x³/x⁴ above
x = 144/√2 = the length of the edge of a square whose diagonal equals 144
(x/6)⁴ = 82944
x⁴ = 107495424
The latter is the volume of the absolute dead center hypercube. The mathematical beauty and harmony of the equation consists in taking the ratio of volume and surface – dividing volume by surface – measuring volume through surface – and demonstrating its perfect identity with hypercubicity through the three unique numbers (and their corresponding hypercubes) which are their own integral products.
The surface area of any four dimensional hypercube equals eight cubes of its one dimensional edge. The volume / surface ratio of any cube is the length of its edge divided by the number of its faces. Whereas it is uniquely the case for the 6 cube that its volume / surface ratio equals unity, it is uniquely the case for that cube whose face diagonal equals 144 that the four dimensional hypercube of its volume / surface ratio is to its volume as the surface area of its four dimensional hypercube is to the volume of that hypercube
The ratio of the volume of the absolute dead center hypercube to its integral product equals
x⁴ ÷ i(x⁴) =
107495424 ÷ ( 1·7·9 ·4·4)² = 10368² ÷ 1008² = 107495424 ÷ 1016064 = 82944 ÷ 784
Leahy states in Measure Beyond Beyond Reach that it is uniquely the case that the ratio of the absolute dead center hypercube and its integral product equals 82944 ÷ 784 ÷ 1 – there being no ratio of any other number in the infinite series of natural numbers with its integral product which equals it. Although I have not examined his proof for the latter theorem to the degree necessary to judge its truth value, I did write a software program in CUDA five plus years ago in which I took advantage of the 384 cores on the Nvidia GPU of the laptop to parallel compute the ratio of all numbers from one to one trillion with their integral products - and I can verify that at least through one trillion no such ratio exists except for the one above.
One of the correspondents of David Leahy was the mathematician John Conway. Whether influenced by Leahy or not he and N. Sloane, the founder of Integer Sequences Online, calculated the Fixed points of the map m -> powerback(m) yielding the series
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 25, 107495424
They state (oeis.org/A131571) "Probably there are no other terms. There are no other terms below 10¹⁰⁰" They give two examples of how to determine the fixed points of the powerback map. To calculate the fixed points (analogous to the fixed points of the integral product series 1, 784, 82944 mapped to the natural numbers) of the powerback series you take the alternate digits of a number from right to left raising the rightmost number to the power of the number to the left of it such that
25 ⇒ ²5 ⇒ 5² = 25
107495424 ⇒ 1⁰7⁴9⁵4²4 ⇒ 4²·4⁵·9⁴·7⁰·1 = 107495424
Now as to whether the three unique numbers which are their own integral products truly do uniquely lie at the foundation of existence and are fundamental intelligible principles which determine the creation of the universe now and in the beginning, or not, the claim of their uniqueness amongst the infinity of numbers appears to assert a real and genuine difference with the literal sense of the second clause of the proposition of AL 4.
Even if these numbers are uniquely universally determinative (and to the best of my knowledge there is no scientific consensus that they are) the question remains as to whether there is any practical, contemplative, or theoretical use for them (with apologies to the Useless Sacred Trees of Chuang Tzu who perfectly survived being cut down by actually being useless to any who might seek to make coffins, doors, and boats of them, either by reason of being too gnarled and full of knots so as to be impossible to measure accurately, or because their branches are too twisted to measure with compass and square, or that the tree passes itself off as a sacred tree to protect itself from those who would hurt it (eheart.com/TAO/CTchapters-small.pdf))
I can attest that there is some small value to these numbers both geometrically and astronomically in regards to the Great Pyramid (a fact discovered not too long ago) – which pyramid itself admittedly is another apparently perfectly useless object (unless of course Khufu actually did ascend to immortal solar or stellar existence through its megalithic technology – and is living there now – or if my hypothesis is correct that the monument embodies the mathematical, geometrical, geodetic, and astronomical knowledge of the Fourth Dynasty, the age at which the civilization of ancient Egypt most likely attained its mensural apogee.
But even if the hypothesis is proved true, since that science remained hidden in that tomb until this day, while science itself advanced light years beyond it, it did not advance science, even if it could have. It may be that the Babylonian and Greek mathematicians and astronomers learned from ancient Egypt, and borrowed freely from them. It may be that transnational communication among the priests and savants of those ancient times was greater than that which has been thought hitherto. Certainly Greeks attested in their writings to the wisdom of Egypt, and many a Greek – be she a geometer or a geodetician – lived and studied in Egypt. It may be papyrus records were destroyed and lost during the long and slow decline of the Library of Alexandria down to Roman times, when (it is said), sections (or at least warehouses) of the Library were destroyed in a conflagration lit at the docks during one of Caesar’s campaigns (en.wikipedia.org/wiki/Library_of_Alexandria), and when the Christians came into power after Constantine, many of the more ignorant, fanatic, and zealous amongst them methodically set about leaving no trace of classical learning, for reasons of faith, superstition, envy, xenophobia, and other ineradicable psycho-pathologies of the human condition (nytimes.com/2018/06/08/books/review/catherine-nixey-darkening-age.html). Still we may content ourselves with the bare fact that a geometrical analysis of the Horizon may recover evidence of their art, and rewrite the history of mathematics, and the science of ancient Egypt, and we are fortunate, living as many a thousand miles from it as we do fom the heep of rubble, in possessing multiple interior and exterior surveys each accurate to within one half of an inch upon which to base the analysis.
It will not, however, solve the aging of the genome, nor correct it to extend life to a more reasonable span in which an individual can actually learn and accomplish something, rather than the short and brief time nature has alotted each of us. So get to children. There is genetic code work to be done.
But even if the Pyramid is the perfect pharaonic ressurrection machine, there is always a Moses to contend with who just might be ordered to anull its quarried causality through the supreme supersession of divine causality in the form of an all serpent devouring serpent staff, and if indeed a Mummy is a necessity to ensure the Ku/Ka/Ba coheres in eternity, a single hungry tomb robber desperately searching for mortuary bric-a-brac to steal in the stealth of night might – in his haste to get in and get out - tear your mummy to shreds, trampling it to dust under foot – and ruin your day in eternity
It even appears these numbers are tangentially helpful in regards to resolving the literal Qabalah of AL (not that working at Crowley's perfectly useless conundrums is a wise or useful way to whittle down your time).
First let us return to the number 331,776 you arrived at by multiplying the first eight numbers of the cipher of AL with each other.
4·6·3·8·2·4·3·24 = 24⁴ = 331,776
It equals the third recursion of one of the Chaldean Oracles cited from the Theology of Plato by Proclus concerning triads - but we can read it simply as All things τα παντα are measured and bounded μετρουν και αφοριζον as the oracle speaks ως τα λογια φησι (as the logos thinks, speaks, affirms, writes) (universaltheosophy.com/articles/taylor/collection-of-the-chaldean-oracles)
τα παντα μετρουν και αφοριζον ως τα λογια φησι = 331776
The number 331776 is perfectly related to 82944 through the equation for the geometric figure of a sphere
x² + y² + z² = 82944
which when solved yields a sphere of radius √82944, diameter 24², circumference 24²π, and surface area 24⁴π = 331776π
If we transform the nine numbers of the cipher into the eleven numerals of which they are comprised
4·6·3·8·2·4·3·2·4·8·9 = 24⁵
their product equals the fifth power of 24, recalling the sole parenthesized verse of the second chapter (if we do not count the 'parenthesized' numbers of II 76)
(This is of the 4: there is a fifth who is invisible, & therein am I as a babe in an egg. )
If we multiply the first five numbers of the second chapter, following the example of the powerback map using a powerforward map, by taking advantage of the fact that in both the manuscript and the 1907 typescript the first four numbers of the cipher are positioned alternately low and high, with the six and the eight written/typed higher than the four and the three, we may take the liberty of reading them as exponents of the numbers which precede them, and lo! the first five numbers of the second chapter derive the volume of the absolute dead center hypercube
4·4⁶·3⁸ = 107495424
Similarly, if we treat the first four numbers of the cipher as one number and the other five numbers as the seven numerals of which they are composed, so that nine numbers of the cipher become "eight and one in eight," the product of these eight numbers divided by the sum of the 19 letters (19=Prime number 8) equals the area of the absolute dead center cube
82944 = 4638·2·4·3·2·4·8·9 ÷ ABKALGMORYXRPSTOVAL
Although I have not done an exhaustive analysis of the rational, linear, and integral products of AL, or any other writings which use the Latin alphabet, and Leahy himself only applied the techniques to Hebrew, Greek, and numbers, and not to any alphabetic schemas mapping living alphabets to ancient origins, I did discover that the rational product of the last clause of the first sentence of III 34 equals
"and the double wanded one assume my throne and place" = 82944
The rational product of I am above you and in you equals 1
I also recursively parsed the entire text of AL out to the number of times at which no word or letter could possibly equal the number 82944, and truth be told, the only eminently sensical word, phrase, sentence, or passage which equals the hypercubic logic of this number of the novitas mentis without being chopped up, lexically broken midstream, or left with a thought dangling in the air is
He shall behold them = 82944
The spelling of the letters of this clause of I 55 are recursed He times – spelled in full to the fifth level of ה
In the identity which equates the ratio of the absolute dead center hypercube and its integral product and the ratio of the only two numbers other than unity which are their own integral products
107495424 / 1016064 = 82944 / 784
the 82944 absolute dead center cube is composed of six faces and two interior measuring planes each of area 10368, and the analogous 784 cube is composed of six faces and two interior measuring planes each of area 98. The product of the areas of one face of each of these cubes - 10368 and 98 - equals the integral product of the absolute dead center hypercube – such that hypercube of the geometric mean of the edge lengths of the 82944 and the 784 cube equals the absolute dead center hypercube integral product hypercube
√ ( √ 10368 √ 98 ) ⁴ = <e>i(107495424) = 1016064
It recalls the King chamber hypercube of the ratio of the geometric mean of its average floor and ceiling height above the base of the pyramid with the length of the chamber: it equals the length of the sidereal year accurate to five seconds. Where H is the chamber floor ceiling geometric mean height and L is its length
(H/L)⁴ = 365.2564 hypercubic king chamber cubits ~ 365.25636 = number of days in sidereal year
Similarly, the ratio of the absolute dead center hypercube and the 82944 hypercube, and the ratio of the absolute dead center integral product hypercube and the 784 cube, each equals the volume of the perfect hypercube
107495424 / 82944 = 1016064 / 784 = 1296 = 6⁴
The twenty edges of the eight planes of the cube whose interior and exterior area equals 8 times 98 = 784, have a total length of 280 divided by the square root of 2 – in other words – they equal the edge of a square whose diagonal is 280. Since the height of the Great Pyramid equals 280 cubits – we can conceive of the number 784 = 28² = the square of the number of digits in one cubit, and its hypercubic logic, as occupying a mean between unity and the hypercubic logic of the absolute dead center cube
6 is the first perfect number (a number equally the sum and product of its factors), and 28 is the second perfect number.
I would not want anyone to jump to conclusions in regards to these numbers. Even though is possible to erect a quasi-consistent system of thought on almost any idea consistency in and of itself is no guarantee of truth – the premises of a system may be false – or if it is empirical the inductions drawn from the repetitions of its observed regularities may erroneously infer patterns from too few instances, or generalize to classes of phenomena to which they do not apply – and so on.
Our original master of magnetism – William Gilbert – notwithstanding his excellent experiments and demonstrations of the laws of magnetic phenomena leapt beyond the evidence to ascribe the rotation and motion of the heavenly bodies, and many another aetiology of nature to magnetism, all the while denying the unity of the forces of magnetism and electricity Maxwell later demonstrated. His enthusiam for magnetism as universal causal cure all was such that it led Francis Bacon to write “The Alchemists have made a philosophy out of a few experiments of the furnace and Gilbert our countryman hath made a philosophy out of observations of the lodestone. [Gilbert] has himself become a magnet; that is, he has ascribed too many things to that force and built a ship out of a shell.”
I will end with a graphic of the triple logic cube, which is simply three cubes of the unitary structure of the absolute dead center cube united through their trinary diagonal logic –
Taking the number of non-stop characters of the Book of the Law as calculated by A.M.O. and independently verified – 23272 – and adding to it one for the circle squared character, there are a total of
23273 non-stop characters = 17 × 37² = number of interior diagonals of triple logic cube × total number of diagonals of triple logic cube²
The triple logic cube is one of those artifacts which – outside of geometrical and logical beauty – appear to have no value – scientific practical or utilitarian. I draw attention to it because of the harmony of its trinary logic diagonal structure with the prime factors of the circle squared non-stop character count of the AL – but like the hypercubic identities above – and isopsephia in general – these harmonies although in one respect beautiful in their own right – are purely anecdotal – and do not imply either prophecy or science.
To explain fully the triple logic cube would require giving an account of the trinary logic at its foundation – I will limit my remarks to the observation that its three logical digits – zero one and unum (the latter depicted by a zero with an overline). Zero does not retain its usual binary logic significance in this logic, both because there is no nothingness in the logic, and in a true trinary logic no logical digit or element of the logical trinity is conceivable independent of its trinary unity – so that the foundation of the truth table of real trinary logic is
O = Ō1
Ō = O1
1 = OŌ
OO = ŌŌ = 11 = ŌO = 1
The alternation of these digits on an infinite two dimensional lattice repeat a three by three pattern – the square essence of which is the foundation stone of real trinary logic – and the top left to bottom right diagonal of that square is composed of three ones – 111 – one one one. There is an analogy here with the least prime magic square which is also a three by three square – its magic constant (the sum of any row, column, or diagonal) equals 111 – its center number is 37 – and the hypercube of the ratio of the first eigenvalue of its matrix with its second or third eigenvalue equals (where 36 √2 is the inradius of the absolute dead center cube and midradius of its face)
(λ₁ ÷ λ₂)⁴ = (111 ÷ (36 √2))⁴ = ( λ₁ ÷ λ₃)⁴ = (111 ÷ (-36 √2))⁴ = 37⁴ ÷ 82944
In the triple logic cube above these (color coded) logical digits are positioned according to the alternating placement of the digits on the interior and surface planes of the cubes, and diagonals are drawn through every occurrence of three identical digits in a straight line. This trinary diagonal logic procedure gives rise to 17 interior diagonals, and two pairs of 10 surface diagonals each – for a total of 37 diagonals – 17 interior and 20 surface – and which include all possible surface and interior body diagonals of each cube, each pair of cubes, and the triple cube.
Just two days after the equinox, this- spring, i tell you, spring!
Thank you for the (sprightly springly?) sage advice!
(I have refraining posting this because it does need to be edited)
Does it still need to be edited, therefore? For those of us who are not mathematically adroit is it at all remotely possible to condense what you're driving at here down into say three or four relatively succinct, manageable paragraphs?
I have limited the account to bare essentials – but the post is still quite lengthy.
You don't say! It must be not far off some sort of a record, and surprising you didn't run up against the computer-says-no barrier for the maximum limit of words postable. How long would the unexpurgated version be, as compared to these "bare essentials"? Two, three, four times as long - or even more?
It may be papyrus records were destroyed and lost during the long and slow decline of the Library of Alexandria down to Roman times, when (it is said), sections (or at least warehouses) of the Library were destroyed in a conflagration lit at the docks during one of Caesar’s campaigns, and when the Christians came into power after Constantine, many of the more ignorant, fanatic, and zealous amongst them methodically set about leaving no trace of classical learning, for reasons of faith, superstition, envy, xenophobia, and other ineradicable psycho-pathologies of the human condition.
Indeed so. As someone or other once said, 'against stupidity the gods themselves wrestle in vain.' The iconoclastic wreckage which was wrought by ISIS upon monuments in Iraq and elsewhere was nothing new at all (though I imagine few would like to contemplate that even those apparently altruistic, 'meek' Christians did exactly the same sort of thing with the Incas & all other primitive 'savage' cultures earlier as well right up to the 20th century. Not forgetting genocide, murder, cursing, torture & and sundry other atrocities alongside the wholesale destruction of artifacts, of course.)
Incidentally why would the 'Writer' board be the optimum location for such a numerical subject as this thread, when ideally it seems like it should best be shoehorned in somewhere with all the other 'solutions' to II.76, etc.?
Norma N Joy Conquest
"Just two days after the equinox, this– spring, i tell you, spring!"
I'm beginning to think there is something to this "Spring is in the air" theory.
Personally, I think the II:76 cypher is a red herring.
Thank you Jamie for your condensation of jg’s post to a version understandable .
Maybe jg can cut down on a while and make a long story short or give us the joke retrenched .
I will let the mathematicians do their complications and the philosophers cut and dry .
“ remember, o chosen one, to be me; “
and i just plain and simple will aspire
“ to follow the love of Nu in the star-lit heaven; to look forth upon men, to tell them this glad word. “
Maybe jg is an Aquarius that can't make sense out of people that don't make sense out a his sense that he thinks he has that others have .