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A mathematical, yet, 2D artistic approach

Towards explorations [2008, 2009] into our ancient past, world of mystery,

Prime & Pattern Numbers  , sacred and/or “natural” geometry, plus more…All seeking..:

Forgotten knowledge and or evidence of advanced knowledge lost in our past; yet, buried in 2D plane sites!!

500x500Khufu554-756-960488PhiRaMid_Spiral_ART555x555-45s-Khufu554-756-960

 

                         

Introduction:

This re-search is focused on explorations and analysis of the mathematical systems employed via

the ancients as evidenced, via texts, symbols, art and or structures in order to

extract lost knowledge and join cross culture metrology paving a connection through our past.

Currently, with the help of University of Cairo Professor Assem Deif, 2DCode has introduced two foundational

Geometrical models (“Tri2Pi” & “Square2Pi”) that bind shape with number for a pure mathematical “natural” harmony….

These relational models (2:1 & sqrt2:1)  have a trail of metrology through our past via Masonic symbols

and structures {ex: Grand Lodge, Rosslyn Chapel};Paul Martyn-Smith finds evidence within areas of his

study of the I-Ching & Tao and their symbols, We find evidence within Euclid’s Prop 47, Pythagorean Theorem,

the Aztec/Maya Art , AE’s EMP/RMP, Integral units of GP, GP overhead view along with the 2:1 GP KC seked.

These sekeds were sacred to the AE... Why?

This site will endeavor to explain why as well as introduce both 2:1 pure mathematical models demonstrated via applied mathematics using metrology ratios of the cultures in focus.

These mathematical models could have real world application within the fields of engineering, mathematics in general, physics and more.

 

Additionally, this research builds off of past pioneers evidenced data and through current collaboration and analysis of the “GHMB Flow Gang”, will endeavor to explore

These natural shape concepts through geometry and pure math to correlate with reverse engineered evidence left behind.

Buckle up and prepare to gain mathematical insight to forgotten knowledge.

 

2DCode-r-past.com conveys no conclusions be made on anything unless provable through solid evidence and or math..

 

Discussions of material have been linked where possible and when time has permitted to actually “find” material posted

**The term “Pi” is used quite often in place of the term “circle” within this data. BASE CIRCLE [D] of unit 1 will equate a circumference [C] of Pi.

Additionally, 2Dcode will endeavor to prove the ancients reconciled the circle via the square. We can gauge both circle and square from one tangent [D].

 

All Hyperbolic theorems, methodology and foundational techniques are defined on the geometry page:

REF: Foundational SQRT2/1:1:1/SQRT2 “Square2Pi” & SQRT3/2:1:1/SQRT3 “Tri2Pi” Hyperbolic Geometry.

Theorems have been formulated and presented as a direct result via these observations and explorations:

- [BASE]x(Pi/SQRT8)                 = SQRT2 PARENT (Square2Pi) “natural” Hyperbolic Side …         [D]:[S]  =(SQRT2:1)

- [BASE]x(2/SQRT3)x(Pi/3)        = 2/SQRT3 PARENT (Tri2Pi) “natural” Hyperbolic Side …             [D]:[TS] =(2/SQRT3:1)

- [BASE]x(Pi/3)                         = 2:1 PARENT:PARENT or  “2:1 natural” Hyperbolic Side …         [D]:[S]   =(2:1)(applies to both Tri2Pi & Square2Pi Geom Models)

- [BASE] x(Pi/4)                        = Hyperbolic Squared (CIRCLE PERIMETER) Side

- [BASE] xSQRT(Pi/4)                = Hyperbolic Squared (CIRCLE AREA) Side

-1/2[BASE]x(Phi^2)                  = Phyperbolic Side

 

Rational 2:1 Geom Model Evidenced within EMP/RMP [AE’s rational sqrt2/1:1:1/sqrt2 model]Discussion…: GHMB | Ma’at

 

“The Rosetta Stone” to the Ancient Egyptian’s Geometry skills (AE 1st Dynasty Hemaka Disks). (Fall ’09)

 

DISC_TEMPLATE.gifJob_B-Egypt_2004_Cairo_122.gifHemaka-Disk3-BASE-TEMPLATE.gif

Photos provided by and copyright Jon Bodsworth 2009- ("The Egypt Archive": http://www.egyptarchive.co.uk )

(many thanks to Jon B. for the best AE pictures available on the web.)

 

Correlating metrology cross culture(AE:Aztec&Maya)

Aztec/Maya World Picture Geometry {joining metrology systems and linking the GP}[May/June 2009]

 

Expanded EMP/RMP study binding the integral units of the GP structure…

Ancient Egyptian Metrology through analysis of Hieroglyphs, GP, Royal Cubit & EMP/RMP or Ahmes Papyrus [summer ‘09]

 

Exploring Ancient Egyptian Metrology through analysis of the Royal Cubit [early ‘09]

 

Attempting to bridge the gap towards a new foundation; joining camps via proof the ancients not only knew, but, embedded sacred knowledge within their structures.

Theorem to Square the Circle’s AREA & PERIMETER. [June ‘09][Posted @ GHMB]

 

Combining Fundamental Hyperbolic Models into one pure mathematical harmony of natural 2:1 Geom Models: [Nov 2009]

The GP Khufu stands at the hyperbolic arc of the [D] diameter ... which splits the circle into two mirrored halves or hemispheres.
hence Pi/2 is the hyperbolic arc formula for a diameter of a circle creating the hyperbolic length unit of one hemisphere... x2 completes the
circle or circumference and relates the diameter via Pi/2 or to the ancients,  11/7 & 7/11, evidenced via EMP/RMP  and confirmed via GP BASE : GP Height ratio with a slope angle

Or 14/11 seked equal to the rational approximation for the Pi/4 hyperbolic arc theorem that squares the circle. Confirmed via GP Height [S]x(14/11)…x4=PERIM that squares GP Apothem [D]x[Pi]=[C]. Additionally, ½ GP BASE SIDE [S] Squares PERIM with GP Height [D]…etc..etc.



833x962-CubeofTime.jpg

 

Exploration of the “Canon” through Harmony of the spheres:

Detailed Demonstration of the “Star of Ishtar” using the natural Harmony of spheres. [Sept 2009 Research]

 

GwheelPenta.jpg   Hyperbolic_864.jpg

 

 

REF LINK: Giza Explorations:    (GHMB Discussion) [April / May 2008]

 

thumb_GPMP_inch_template

 

Sacred and or Natural Geometry Page: Here (Defines hyperbolic theorems and current techniques)

 

Convergence:

{August/Sept 2008 -} Converging Pi, SQRT’s  and more with potential output pattern in an approximation of the same input…[Posted @ GHMB]

 

[July/August 2008] How much knowledge did the Ancients really have? And, how much did they “intend” to convey or even hide in “plane” site 4 that matter?

For example…let's take the Giza Plateau that John Legon and others convey a root2 x root3 Giza-Rectangle with a root5 diagonal!

But, as an “approximation” to what we currently call true SQRT2 & SQRT3.
If these roots were intended and can be proven; then, how could the Architects not also have known the combined approximation of Pi?

SQRT2 (99/70)+SQRT3 (97/56) approximates to Pi!

Below, is the geometry / ["art"] that binds the two "roots" of the square into one dual harmony of BASE 360 geometry....! :)

http://www.2dcode-r-past.com/Geometry/PI_SQRT2-SQRT3.gif
posted [GHMB Aug 2008]

Could the Architect of the GP’s have known SQRT2/1:1:1/SQRT & SQRT3/2:1:1/SQRT3 via approximations?

Furthermore, could Ahmes have used a BASE 360 degree system similar to the above root2 & root3 diagram? And, can we find signs in the RMP?

The AE used fractions and could easily approximate these root approximations.
(99/70)+*(97/56) = 3.1464285714 =~ an approximation of Pi {true Pi= 4xATAN(1) using radians}.

[Dec 2009] After two years of exhausting research, I came upon the 1st Dynasty Hemaka Geometry Disk excavated by Emery published in 1938.

Hence, It would appear the answer to “if” the ancients understood basic shape models and root approximations can be concluded “yes”.

Please observe:

840-Hemaka-Chessboard-disk.gif

Posted [GHMB Dec 2009]

Plato also mentions: “…. root 2 as the irrational and rational diagonals of 5”

5*sqrt(2) = 7.0710678118... = [10/SQRT(2)]
therefore,
1/(sqrt(2)*5) = 0.14142135623 = [SQRT(2)/10]
hence,
10/(sqrt(2)*5) = 1.414213562373095 = [SQRT(2)]


And, can be broken down algebraically by dividing the 10 by 5


Therefore,
2/SQRT(2) = 1.414213562373095 OR, 2/SQRT(2)=SQRT(2).
hence,
x/SQRT(x) = SQRT(x)  = “The underlying math from Plato as it was processed (
[via post by Ariston])


Furthermore, if we apply those SQRT2 approximations; used most notably by John Michell, and mentioned many times throughout this site.
99/70 = 1.4142857 cyclic "142857"
140/99 = 1.4141 cyclic "41"

They actually switch values upon 2 as the numerator using (AE style) AE 2/PrimeN fraction table.[Exploration - Expanded study to 999]
2/(140/99) = 1.4142857 cyclic "142857"
2/(99/70) = 1.4141 cyclic "41"

 

And, if we dig a little deeper into the [Wiki]Rhind Mathematical Papyrus

 - 

We can observe indication of intention via these very units in the company of diagonals.

First, we must define a diagonal to understand the relation. Rhind#1 starts right out and sets the foundation with a study of the diagonals!

If a square measures 70 by 70 royal cubits, the diagonals measure practically 99 royal cubits,

and if a square measures 99 by 99 royal cubits, the diagonals measure practically 140 royal cubits

Hence,

[Exploration - RMP#1 SQRT1/1:1:1/SQRT2 Exploration]; Discussion…: GHMB | Ma’at

70/99            = 0.7070

99/70            = 1.4142857142857

140/99          = 1.4141

99/140          = 0.70714285714285

 

SQRT2/1:1:1/SQRT2 (…through rotating root approximation)

(70/99) /(140/99)     =  .5 

(140/99)/(70/99)      =  2

(99/70) /(99/140)     =  2

(99/140)/(99/70)       = .5

 

Proof the AE understood convergence within the Rhind?

Further evidence of SQRT2 convergence within the Rhind.

Proof of SQRT5 convergence within the Rhind?

 

**-- Jim Alison also conveys these very same digits [70,99,140,198,280] via a similar concentric; yet, “square only” concept

…adding more support to the theoretical concept being intended and supporting potential advanced AE knowledge in conveying

a relational “system” of units similar to [Square 2 Pi] Duality that was extracted via Khufu GP overhead.

If nothing else; the data appears compelling enough to pursue further investigations.

 

Visit the explorations page for:

Theoretical deduction of how the cubit rod and string is all that would be needed for the AE to know SQRT2/1:1:1/SQRT2 relational geometry.

 

RMP50

Exploration - Base 9 geometry layer to represent Ahmes 9 unit start area. [posted: GHMB | Ma’at]

Exploration - Base 16 geometry layer to break the RMP50 down to 256.  [posted: GHMB | Ma’at]

 

Interesting diagram of Exploration - 256:81 Pi-tangle from working RMP50 [Instructions How to create] Posted: [GHMB]

 

-          the natural BASE 360 layout of the 256:81 Pi-tangle intersects at the ROOT2 intersection of the square to inner circle ratio..!

-          Also forming a perfect 256 square interior of the base 360 geometry which approximates the ROOT2 Inner square in ratio with BASE.

-          256:360 = .7111 with a reciprocal of 1.40625 exact. So, we have another ROOT approximation extracted from the Rhind/Ahmes Papyrus.

 

RMP 44&45

“If the edge of a cube measures 10 royal cubits or 70 palms or 280 fingers = 5 x 56 fingers,

the diagonal of a face measures practically 99 palms,

the cubic diagonal measures practically 5 x 97 = 485 fingers.

 

SQRT2 APPRX = 99/70 = 1.4142857

SQRT3 APPRX = 97/56 = 1.732142857

 

we can also observe permutations of the Exploration - 6x6 cyclic puzzle cube buried in both root approximations.

 

GP & 22/7 Pi Approximation:

…If we “just” take the current orthodox understandings, and accept the “AE seked” ratio of [14:11]

and employ the inverse or reciprocal (1/14:11) or (11:14) per side angle.

All 4 angles of that Khufu GP, then add up to “exact” 22/7 “natural”… cyclic Old School Pi!

So, one could make a case for the GP representing Pi/4 approximation “just”  by natural rectangle design alone.

That Pi/4 relationship has been demonstrated to be a close approximation for calculating area of circle.

 

Now, back to the question of what the AE really knew? That is the main focus of what can be found in this site and throughout this research.

…Simple coincidences in data are easy to dismiss…but, an abundance of mounting data is hard to ignore when grouped and analyzed together…..

Per chance this site will assist to paint a portion of those coincidences grouped and conveyed through art, geometry & math towards proof that the AE

Were potentially more advanced then current mainstream accepts…

 

After all, shouldn’t we expect to find correlations between the units and methods found in the structures and monuments observed from our ancient past

 with any and all available math data [proof] of their implemented math systems and formulas employed…. in the very same era?

 

Furthermore, will not signs of our current math systems utilized in current structures by current Architects be present thousands of years from now?

**....given any remain in thousands of years from now?

 

….much much more scattered throughout this site.

 

GP Tri explorations collaborating with Derek Skhane and various GHMB researchers’ via a BASE12 design…

- 2DCode GP Khufu Study comparing John Legon’s theoretical shaft measures in correlation to Petrie GP Course Data via D. Skhane BASE12 design.

- Additional Explorations of the GP’s, Giza and AE Math…

 

Additional Explorations & Research documented and or noted for 2008-…

Additional Explorations & Research documented and or noted for 2009-…

 

*Visit Ron Knott’s website for anything you could want to know about Lucas, Fibonacci, Phi, phi and the Golden Mean, Pattern, Sequence etc...

*Jim Alison’s website also demonstrates some interesting work exploring phi and the measures of the units within the structures of Giza and more.

 

 

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