MIRRORS!

Sacred Geometry: Square Roots 2, 3 and 5, pi and phi.  That's pretty much it - unless you include the 5 Platonic Solids!  So, draw a square and connect opposite corners ... Root 2.  Draw a cube and connect opposing corners ... Root 3.  Draw two squares side-by-side and connect opposing corners ... Root 5.  And Root 5 ties to the Golden Mean, the Perfect Proportion, phi (and Phi - lower case indicates 0.618 ... and upper case 1.618 ... - they're reciprocals/mirrors of each other).

Phi can be roughly tied to Pi, the relationship between a circle's diameter and its circumference: Pi = Phi Sq. X 6/5-ish.  But it's the 'ish' that's important.  Geometry can draw things (theoretically) perfectly that numbers cannot perfectly express.  The numbers are approximate representations - and that applies to Roots 2, 3, and 5 as well as phi and pi:                    

note - apparently it was once thought that nuts (note the almond-shaped vesica) made you pregnant!

A few years ago, Adrian Gilbert conducted a Midsummer tour of sites of London that demonstrated Sacred Geometry in practice.  He took his companions to the London Monument and Nelson's Column, for instance.  The Monument is 202 feet high.  At a certain time of day on June 21st, Midsummer's Day, it casts a 350* foot shadow.  And the ratio 202 : 350 indicates the Square Root of 3.  It's pretty close.  The Square Root of 3 is the maths of Gothic Cathedrals - half the vesica above is basically a template of the arch design that lets in light their light and helps distribute the weight of the roofs.  Gilbert could draw on obelisk evidence from Egypt to argue that the association of Midsummer with Square Root 3 was informed and intentional (on the part of Wren and Hook, both FRS - my note). You can see more in THE NEW JERUSALEM, Adrian Gilbert, 2002 AD - and note that the London Monument stands in the precise area that Sir John Evelyn, also FRS, planned to reconstruct (post-Great Fire) London to a Kabbalistic 'Tree of Life' design (click on Geometry/Diags. for his plan et al).  

*Taken to be 12 lunations by some, elsewhere.

Unfortunately, the day's session with Gilbert was (sadly) somewhat marred: he applied exactly the same dimensions and interpretation to Nelson's Column, Trafalgar Square: another 202 foot construction with another 350 foot shadow.  'Cept it ain't!  Gilbert - or someone - had double-counted Nelson.  The whole construct was thought to be 185 feet high, including a 17 foot Nelson ... and 185 + 17 = 202.  Recently, even the 185 has been rejected in favour of 169 feet and 5 inches for the whole thing.  Oh, well - but a least Gilbert was (quite rightly, in my opinion) able to point to the significance of the Latitude on which the Monument and Column stood.  51.5 is the value he gives.  'Cept it ain't.  51.5 is the WRONG SIDE OF THE RIVER THAMES.  The correct (and significant) value is 51.51, the mirror of 1515 (Isis) otherwise Thames.  Still, I'm quietly confident I've stuffed up more than once here, too ...

202 : 350 is not the only Square Root 3 approximation - but, as Gilbert quite correctly points out, it does 'live' easiest of the alternatives with a Pi of 22/7.  The ratio most commonly quoted for Sq. rt. 3 is 153 : 265.  And this is where the fun starts.

The vesica piscis above (the Sq. Rt. 3 almond shape) can be expressed as a fish - if you add a little 'tail' fin to one end, that is - and a fish was used to denote Christ, amongst other symbology.  In fact, at ST. JOHN (21, 11 - 21 X 11 = mirror II3II) precisely 7 out of 12 disciples catch 153 fish when spurred on by a resurrected, but unrecognised, Christ.  Well, 12/7 is the simplest Root 3-alike - and also hints at musical scales, the stars above et al.  But 153 : 265 is also a Sq. Rt. 3-alike (as are 'Jesus' plus 'John'/7 - click on Jesus and John for the explanation).  Note that the zodiac sign Pisces has a piscis-y representation (click on Geometry/diags for this and more) and that Christ was on a zodiac cusp:  Aries the Lamb and Pisces the Fish. 

153 is more than just a Sq. Rt. 3 indicator.  It's also the sum of 1-17 inclusive.  Its mirror, 351, is the sum of 1-26.  26 is the number of letters in our alphabet, MM, and the sum of the middle pillar values on the Tree of Life: 1 + 6(sun) + 9(moon) + 10.  351 is what you get when you add R to U and mulitply by 9 - and the Ru is the mirror part of the Egyptian Ankh, and the 'female' part.  Lady Anson of Shugborough, rather rudely and crudely, makes reference to this fish-mirror (and where-a-lady's-legs-meet) symbology when she writes of a "mirror of true reflection".  It's a word-play.

The ratio 153 : 265 can be expressed as 51 X 3 : 53 X 5 - note all the 5s, 3s and 1s (666*/4 X 53 = a pretty useful Pi! And note, too, here that the REVELATION 'Beast' is on top of  the REVELATION 'Bride'). They are thematic in MOVING QUOITS - messin' with Maths, I call it (like discovering that 1 cubed + 5 cubed + 3 cubed = 153). The Michell Decagon (click on Perpetual Choirs) lives fairly nicely between Latitudes 51 and 53.  And 51 is 17 X 3, 17 being the basic number in the Phi serpent, a mobius strip arrangement along which the 'Golden Mean' numbers of the Fibonacci Sequence are distributed.  The Sequence runs (1), 1, 2, 3, 5, 8, 13, 21, 34 (2 X 17), 55, 89, 144 ... Add any two consecutive numbers to find the next, whilst bigger over smaller approximates to Phi whilst smaller over bigger approximates to phi.  Note the 5 : 8, pentagon to octagon. As 15 : 24 this finds expression both in the Biblical REVELATION as the number of elders round the throne of God and in David Wood's GENISIS as 'the only way' (it isn't!) to generate the Isis model (click on Geometry/diags for more on this). 

Note 10-04-08:  666 as the Number of the Beast in doubt.  An early (if not the earliest) MS of REVELATION seems to give 616, otherwise 2cubed X 7 X 11. 

Archimedes knew of 153:265.  Fred O Mills tells me of a mystery: Archimedes write of the Square Root of 3 being a value between 265/153 and 1351/780.  I note all the 1s 3s and 5s again; Fred notes the mystery 'How did he know?'  Thanks for that, Fred.

If we look again at mirrors 153 and 351, note they are 198 apart - a British 'rod', otherwise 16.5 feet and 5.5 (Fibonacci) yards.  16.5 is 15 + 1.5 (Isis).  REVELATION's St. John instructs us to rise and measure the temple of God ... with a 'rod' (furlongs also rate a mention).  198 is a mirror constant as in 123 : 321 and 264 : 462 etc.  If we sum 153 and 351 we get 504.  504 is quite a few things: 63 miles in furlongs, the radius of the (original) Michell Decagon, 1/4 of a Platonic zodiac 'Age' (as in Aries, Pisces and Aquarius), and pretty much 10 X Cube Root 2./2.  Note that Cube Root 2 mirrors Square Root 3.  Returning to 16.5, it is the product of two mirrors that are mirrors themselves: 6.9 and 9.6, sun/moon combinations (of 6 and 9, otherwise basically 2 : 3 and 15 : '666'.  2 : 3 was a basic theme in Plato's Atlantis ground-plan).

Planet Earth wobbles slightly.  The result is that we are said to be 'in' one of twelve zodiac Ages at any given time, moving counterclockwise.  The Platonic canon for the whole cycle is 25920, otherwise 12 X 2160, and the name given to the phenomenon is Precession.  Where the Earth a perfect sphere, there would be no precession (for which a truer, less pretty, value is nearer 25800 years).   Full Platonic precession is 72 X 360, otherwise 72 X 72 X 5 (where both 72 and 5 are pentagonal in connotation).  72 (a pentagonal angle) is the mirror 2 cubed X 3 squared - but then 504 (above) is mirror 2 cubed X 7 X 3 squared as 360 is obviously 2 cubed X 5 X 3 squared.  Note also the number of Saints in a Perpetual Choir, 2400.  That's mirror 2 power 5 X 3 X five squared.  Note how similar 2400 is to (Jesus and John) 7 power 4, 2401.  Note also that the "As Above So Below" mantra and mirror that is destructively evident in Lear's madness and Macbeth's treachery (Shakespeare) has a problem here.  Ideally, an Earth year would be 360 days - but it isn't.  Hasn't stopped some imagining it must once have been.  "As Above So Below" slides into our 'Lord's Prayer' ... (on Earth as it is in Heaven).

In this detail, note the twins.  One is in white pointing up whilst the other, in blue and red, points down.  One holds a Golden Key.  So we have two figures and 2 sets of mirrors (up/down is one, and twins/mirrors is the other).  The up/down finds a reprise on Golden Dawn's Tarot 15 Devil Card (click on Tarot II and scroll down).  The game here is reciprocals.  David Wood, GENISIS, is just one who takes 666 (quite wrongly) to be the reciprocal of 15.  It's actually 666 recurring (if we drop the decimal points).  666 is a combination of mirrors: 234 and 432. 

432 and 666 are said to be solar: the diameter of the Sun in miles is given as 864000 miles (432 X 2000) whilst the Magic Square of the Sun is 6 X 6 = 36 and can generate sum 1-36 = 666, otherwise 74 (Jesus, Lucifer, Tarot, Menorah and He Phren) X 9.  864000 is the product of two mirrors: 2400 X 360.  It is the British measure, the mile, that makes these fit.  But a mile is a mirror: 063 to 360 = 63360 inches.  Now, were a British inch any longer or shorter, this wouldn't work.   Note that 2400 : 360 is a 15 : '666' relationship and also note that whereas 864000 is solar, so 2160 is lunar (and 3960 associated with Planet Earth).  All these numbers are from the decagonal 36 X table: 36, 72, 108, 144, 180, 216, 252, 288, 314, 396 .. 504... 792 ... 864 ... and they are all '9' (3 + 6 = 9, 2 + 8 + 8 = 18 and 1 + 8 = 9 etc).  I've read somewhere that 9 was the number of the twin and lunar goddess Artemis when she walked on the Earth.  Many of the numbers above are called "magnificent" by Bonnie Gaunt in the context of one of our Perpetual Choirs, Stonehenge - she claims to have found them evidenced there.  72, 144 and 288, I read, were basic to Platonic Atlantis - John Michell. 

666 can be seen as 18 (6 + 6 + 6), the sides of an equilateral, and precessional and decagonal 2160 (6 X 6 X 6) as well as the stylised reciprocal of 15.  It is also a Venus at dice: 6 6 6.  Its mirror is a 'dog', a Canis, 1 1 1.  The mirrors sum to (extravirginal) 7 7 7.  All opposite dice faces sum to 7.  And dice also are one type of the 5 Platonic solids, cubes.  They carry 153 and 351 on their faces as well as 153 and 264 - remember 153 : 265 Sq. Rt. 3.

Turning to Pi, useful approximations are 22/7, 223/71, 666/212* (the REVELATION beast 'over' the REVELATION Bride) and 355/113 - this last being correct to 6 decimal places!  22/7, and 355/113, at least, originate in China circa C5th AD.  Earlier attempts at quantification involved hexagons.  You put one outside and one inside, both touching the circumference.  These provide an outer and inner limit for Pi, relative to the circle diameter.  Obviously, the more-sided the regular polygon, the smaller the range.  Archimedes (C3rd BC) used this method allied to Square Root 3 approximates to derive a range for Pi of 3 10/70 - 3 10/77 (where 3 10/70 aka 3 1/7 is 'schoolboy' 22/7).  Other values used have included 3 (the Biblical .'Molten Sea' details), 3.125 (Babylonians C19-C17th BC) and the range 3.11111r-3.16.  Sir William Jones (1675-1749), FRS, with links to the FRS Parkers of Shirburn and the serial FRS Yorkes, invented the use of the Greek symbol for Pi in 1706 AD, possibly inspired by William Oughtred's work (lived 1574-1660). 

*666/245 approximates to "the natural logarithm", e.

These 5 then, Square Roots 2, 3 and 5 together with phi and Pi, consitute much of Sacred Geometry - and we've met with the cube as one of the 5 Platonic Solids.  Now I think it's 'sacred' because it applies to the maths of sacred/holy buildings; others perhaps see the hand of the Great Architect and Geometer and consider the naths itself sacred.  Both interpretations are possible.  What we need to address next is why 360 degrees describes a circle.  The value is, after all, utterly arbitrary; it could just as easily be 361 or 365.25 or 6.9823.  The answer to why 360 is factorisation.  The numbers 1, 2, 3, 4, 5, 6, 8, 9, 12 all divide into it.  Only 7 and 11 (note Pi!) don't - and 7 is termed 'virgin' for this very reason, it is not possible to perfectly quantify 360/7.

Before leaving mirrors, I reprise the mirror symmetries that are regular polygons, their various stars (-agrams), and, of course, circle, sphere and Platonic Solids.  We've seen how 'Adam' plus 'Eve' = hexagram and how 'Lady' plus 'Goat' = decagram.  Similarly, rotate a square to get octagram, and so on.  This mirror theme also applies to diamond/lozenge/rhombus - Adam cleaving to Eve, and a shape implicit in a vesica.  Whirled through the air in a lemniscate figure 8 these were called 'Thunderer', a name applied to John the Evangelist and his brother.  The pattern fits to St. Mary's Chapel, Glastonbury, and to Stonehenge, where a 7 to 12 relationship can be found (see Michell/Street for this, and John Michell's THE TRAVELLER'S GUIDE TO SACRED ENGLAND for the St. Mary Chapel floorplan - www.gothicimage.co.uk/books/sacredengland2.html).  Michell notes the width of St. Mary's Chapel as 39.6 feet and its acreage as 0.144 acres - both are 36 X table (and 'magnificent') numbers familiar from above.  But there's one other thing, t.

The tetrahedral constant, t, is the angle from the centre of a hexagram to a vertex.  It also applies to a derivative of two of the 5 Platonic Solids, the key being the 4-sided pyramid.  Its value is roughly 19.5 degrees, 2 X which = roughly 39 degrees.  In a right-angled triangle, that would generate 51 degrees.  And that's pretty much where MOVING QUOITS comes in: 396 (198), 144, 51 and 39 - the numbers used in the construction of the REVELATION 12, 1 Woman, Isis - with 33, 3 JEREMIAH thrown in for good measure.  *MOVING QUOITS is my manuscript title.  The pages published here are background research notes for that - necessary study but not necessarily included there.  For example, tetrahedron: ratio 1 : 2.720 ish to sphere enclosing it (surface areas) = Thom's posited MY (Megalithic Yard).

THE KEY MIRROR in the Michell Decagon/Perpetual Choirs model is a real gem:  (2 cubed X 11 X 3)squared.  And that begets 69696.   You "read" it 2 3 11 3 2 and this comprises mirrors 231 and 132.  Now, 2 132s are 264 .... and 264 squared is ..... 69696.  Neat!  231 + 132 = 333 (akraisia, to some), half of 666 and evocative of 33, 3 (Jeremiah above, and where 33 X 3 = 99, where 231 - 132 = ....).  69696 is currently unique: it is the only known "palindromic" square (aka 'twin prime pair sum').  666, note, is the mirror 2 X 37 X 3 squared.  (Rem. basic 6 : 9 aka 2 X 3 : 3 squared).

264 is 2/3 of John Michell's original 396 CPC circumference (1972) and, like 396, is a number called 'magnificent' by American writer Bonnie Gaunt (see above) in the context of Stonehenge and the Great Pyramid, Giza (noting. too, Michell on Glastonbury St. Mary's Chapel, above).  Another relation is equally 'magnificent' 316.8.  But then it's ALL of it pretty much a circular argument, anyway ...

!CAUTIONARY NOTES!: current mathematical notations support much of the above 'fun' - if you look at just the numbers and use say 2*3 for 2 cubed.  The problem is that whilst the above is enticing, is it actually possible (other than comparatively recently)?  When does the convention for little airborne 2 (for square) and 3 (for cube) etc. actually 'kick in'?  I have found the notation in both Newton (1642-1727) FRS and Leibniz (1646-1716) FRS.  I have NOT found it in Oughtred (1574-1660).  dr.math@mathforum.org tells me of a FAQ section at the dr.math website.  There I find Rene Descartes was the first to use raised numbers for integer powers (1637).  At the same site I find . and * both dated as X signs/alternatives.  Oughtred (above) introduced X whilst the dot was used by Harriot (1631) and the * was used elsewhere (1659).  A raised x power existed in 1484.  Log tables incl sine, cos and tan appeared end C17th AD/early C18th.  Have some fun: search them for the values that generate 0.618 and 1.732 etc. ...

CAUTIONARY NOTES: Mirrors like Lat 51.51 and River ISIS are somewhat enticing, don't you think?  Problem is, the numbers 5 and 1 (as with the rest of the Hindu-Arabic 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 system, only arrived in the West late C10th AD, were widely used ?by (surely ?from) the time of Fibonacci, 1202 AD, were first known to be used in England  C15th AD to become the 'most commonly used' system (Wikipedia) by the mid C16th AD - but the identification of the River Thames as the River Isis dates to 1540 AD (Leland wrote of : 'New Bridge on Isis') whilst the identification of the River Thames with 'Thames' is post 1599 AD!  In the C14th AD the river was 'Isa' in some of its documentation, and Isis generally denotes Oxford to source.  Some see the 'Isis' as older Tamesis shortened, I note, although the numbers themselves derive from elsewhere and the C7th AD and beyond.  The decimal point (51.51) dates from 4 and 1/2 thousand years ago but wasn't really an adopted convention in England until the mid 16th - early C17th AD.  "51.51" didn't exist to play games with (in this country - at least exoterically) much before the Restoration it would appear!  This CAVEAT also applies to Bible Chapter and Verse 'fun' - doesn't apply in Roman Numerals!  Similarly things like 666 become letters - DCLVI - and games with mirrors involving powers only apply in (recent) H-A: the notation wasn't available previously!!!!

I do well to worry: the Biblical mirror possibilities created by H-A are 1661 AD Old Testament earliest (c/o Jan Leusden) with NT?  Later?!  Quite possibly.  Originally there was Hebrew notation.  Early C13th AD Stephen Langton introduced 'Chapter', then letters A-G and/or A-D were used to group 'verses' in 7s and/or 4s, sometimes run alongside other notations ... 

I'm not happy with the above note: Gerbert (Pope Sylvester II) was acquainted with the Hindu-Arabic notation 980 AD (see below) and Fibonacci (circa 1880-1250) used it in his LIBER ABACI, 1202 AD.  But it's I/Johannes de Sacrobosco, a contemporary of Fibonacci (1195-1256) I'm unsettled about.  This guy wrote the basic - and widely used - navigation textbook TRACTATUS DE SPHAERA, 1220 AD.  Now this astronomy book was used until the early C17th AD - besides demonstrating various proofs that the Earth is round.  And I read that I/Johannes ALSO employed the Arabic methods of Arithmetic and Algebra.  What's more, he's English and becomes the first astronomy book writer to be printed (as against copied by hand etc).  And the date for that?  1472 AD - well before Oughtred, for instance.  Wouldn't works as famous as these be discussed - and even sought - by academics, navigators and/or cartographers?  More to follow on this as I get replies from Academia* in ... but Michell's CPC (click Perpetual Choirs) is a game played out on Latitudes with a centre just by Hollybush.  I/Johannes de Sacobosco was, otherwise, I/John Holywood or Holybush**.  "Hollybush" arrived on maps some time after 1800 AD. * *(Some refer to him as 'John of Halifax').

*First academic response notes the practical appeal of Fibonacci to Traders (John O'Connor, U. St Andrews ref. Devlin - who dismisses non-Fibonacci sources for H-A as too ivory-towered by far).  But elsewhere I read that De Sphaera was THE required reading for seamen.  This circle doesn't square.  Even more so if you consider John Cabot (Anglecised from the Genoese):  born Genoa, lives Venice, sails from Bristol.  End C15th.  H-A or Roman numeral calculator?  My money is on the former.

I have refs. turned up re arrival of H-A in West 662 AD, + its banning in 1299 Florence (open to fraud!) and being finally established from below - meaning the merchants used it until it became the way-you-go adopted everywhere C15-16th AD (from below/underneath meaning not imposed by Gov./from above).  The actual digit characters we use are called 'Western Numerals' or 'Western Arabic Numerals' (etc.) exampled in 950 AD and spreading from about 1000 AD.  The number 4 would be unrecognisable but 6 and 9 are clear, and  Roger Bacon, say, (circa 1214-1294) appears to have used a version that included the modern 6 and 9.  By the time of printing and the Gutenberg Bible (1454) all the digits are recognisable and they become standardised (by this and from this) those we now use*.  Early for instances in the country are a church tower inscription in Sussex (1445), a church gate in Berkshire (1448), a tomb at Elgin (1470) and a belfry door in Dorset (1487).  Whilst I'm on the subject of H-A (as us cool dudes term Hindu-Arabic!) consider 666.  When REVELATION was written there was no '6' (that I'm aware of).  In Latin: DCLVI, in Greek try exakosiosi exekonta ex or Chi Xi Digamma (XES/or xes - it makes a difference, apparently) , and in Hebrew mem-final samek waw/tav gershayimi (or is it ?n for m - my handwriting's fault!) resh samekh vaw geresh.  Something like that anyway.  But not the symbol '6' as is.

*see http://en.wikipedia.org/wiki/Arabic_numerals (incl. Roger Bacon's) and also, by comparison, http://www.maths-rometus.org/mathematiques/maths-et-nombres/histoire-de-la-numeration.asp

This below is from there and comprises symbols from towards the C10th AD (Chiffres Indiens):

 I've just stumbled across Bill Heidrick's fascinating Gematria site (esp. 'Values' - click on one to find both meanings and maths)
www.billheidrick.com/works/hgemat.htm I see for example 576 (eagle to Derek) given as a value I can mirror: 2power (3 X 2) X 3squared ... Bill has 567 (my noos opposite of he phren) as 3power4 X 7, a seveny thing of sorts, remembering 47 and 74 stuff.

Of note, an email (thanks to Mr. Heidriick):  consder permutations in the context pf 'mirrors' is one bit of advice.  Let's comsider:  Jesus generates 666 (74 X 9) and mirror John generates 423 (47 X 9) where perms of 423 combine to generate 666. We have the mirrors 234 + 432, But we also have 243 +423 and 324 +  342 ... and indeed one of the  153 perm mirrors also generates 666 (135 +531)  and the other perms  can also produce this outcome ( 153 + 513 and 351  + 315).  So thanks for that - and I'm sure other 3 number combinations can also inform the idea!   What do you know: John + his mirror = 747 (423 + 324). It's all like this in this little self-contained mathematical 'world'!  747 = Tehuti (moongod/ wisdom god Thoth).

Note (11-03-08): relating to the above and the Jesus-John 74-47 (121) mirrors.  Consider 12.  That's 3 X 4 (where its esoteric pal is 7, 3 + 4).  The mirror of 12 is 21, 3 X 7.  If you add 12 and 21 you get the 33 some claim masonic significance for and, if you square them, you get 144 and 441.  Now this last, 441 is in a cubic progression of total cubes in cubes divided into smaller cubes such that there's 1, then 2 cubed (8), then 3 cubed (27), then 4 cubed (64), then 5 cubed (125) and so on.equal smaller ones.  First there is a total of 1 cube, then 9 (4 small ones and the overall big one), then 36, then 225 and then 441.  This progression is the square of what happens if you start with a dot, add a second and join them up ( = a straight line), then add a third dot and join (= 3 straight lines) etc.  Oh! Those long TV BC repeat hours to while away ....  37 and mirror 73 are primes 12 and 21 and 3773 is mirrors 2701 + 1072 ....and 3 X 7 X 7 X 3 is 441 (above)  ... and the 37 times table is that of 666 ... and 37, 377 and 11771 are the aberrant numbers on the phi serpent distribution ... and ...