Zep Tepi Mathematics 101 I Douglas 2021 Zep Tepi Mathematics 101 Ian Douglas, B.Sc ian@ zti.co.za 16 January 2021 https://doi.org/10.5281/zenodo.4410650Version 1.1.0 DOI: This wor! is license" under t#e Creati%e Co&&ons Attri(ution 4.0 International License. +nqi": )-a521/-)06)e0c16))11./6a2a0e6(c 2lease c#ec! via DOI for latest version. Abstract ' mathe&atics course fro& t#e Ze4 Te4i era, w#ere we plan an" analyse a large buil"ing site, s#o ing ho the design mirrors the stars. ' si&4le and elegant e64lanation o/ ho Giza, wit# si6 main pyrami"s, was lai" out, using √2, 81, √5, π and φ. The design incor4orates t#e necessary ele&ents for s,uaring t#e circle, area- ise. The design matc#es the hea%ens around 5...! B$E. This coul" force a ret#ink o/ at least t#e #istory o/ mathe&atics, i/ not the broader hu&an ti&eline, an" effecti%ely sol%es the puzzle o/ #o Giza was lai" out. >ey or"s: Egy4tology, 7iza, pyrami"s, alignment, geo&etry, arc#aeogeo&etry, arc#aeoastrono&y, history o/ mat#e&atics, π, pi, φ, gol"en ratio, s,uaring the circle. Best vie e" and printe" in colour. 1 Zep Tepi Mathematics 101 I Douglas 2021 Contents 1. Introduction 2. Notation, accurac and methodology !. The p ramids and their locations ". Module 1# The Thoth Grid %. Module 2# Thoth’s La(, or The s)uare on the h potenuse... *. Module 3# Di+iding space in a gi+en ratio ,. Module 4# The s-eleton blueprint .. Module 5# The big triangle /. Module 6# S)uaring the circle 10. Module ,# A map o1 Khu1u3 11. Conclusion 12. Ac-no(ledgements 1!. Bibliography 1". Gizactor version 3 Updates# 1.1.0 Re-arranged discussion of P1 base size, e:panded discussion of 21"!, e:panded discussion about the big triangle and put it in its o(n section. Added “Map of Khu1u3< section. Assorted other bug-=i:es and embellishments. 2 Zep Tepi Mathematics 101 I Douglas 2021 1. Introduction ?The language o/ Giza is mat#e&atics.@ Ao(ert Bau%al ?Bou will belie%e.@ The arc#itects o/ Giza 5#is is my fiD# an" a hal/ atte&4t at writing this since early 2020, eac# ti&e starting at a di=erent 4oint an" follo ing a different path. As with t#e rest o/ my e64lorations o/ Giza, it has been a cyclical process, one thing leading to another, forcing re%isions and reCine&ents. It feels li!e I #a%e been gui"e" along li!e a c#il" learning science, w#ere w#at t#ey teac# you in junior sc#ool is “correcte"” in high sc#ool, an" then again at uni%ersity le%el. So w#at starte" o= innocently as a quest to see i/ Giza was hi"ing any circles di%i"e" in t#e gol"en ratio has no progresse" to a goo" understan"ing o/ e6actly ho t#e site was lai" out. The design presente" here largely su4erse"es the met#o"s presente" in my earlier paper. The "esign is elegant and de&onstrates undeniable kno le"ge o/ π, φ, an" s,uare roots. The final sur4rise is t#at the design inclu"es s,uaring the circle area; ise, not peri&eter; ise as in K#u/u. There are also nu&erous mat#e&atical “tric!s” or “jo!es” along t#e way. 's suc#, Giza re rites the history o/ mathe&atics, and we nee" to sto4 gi%ing t#e Gree!s cre"it for things that originate" in Egy4t. This paper is a continuation o/ my pre%ious efforts, Dis!er/ery and the Alignment o/ the Four Gain Giza Pyrami"s (Douglas 201- I1JK H#ence/ort# Dis!er/eryK and 5.,..0 B$E and the 21 Stars o/ Giza (Douglas 201- I2JK H#ence/orth ..>K. The first atte&4te" to deter&ine the location o/ the /ourth pyrami", as docu&ente" by Nor"en I1J, w#ile t#e second s#o e" a stellar alignment around 5...! B$E. This docu&ent intro"uces t#e fiD# an" si6th pyrami"s, long de&olis#e", for w#ic# we currently #a%e no conventional e%i"ence. Ho e%er, t#e stellar alignment strongly suggests t#eir presence. Once we a"" the& to t#e layout map, multi4le fascinating mathe&atical relations#i4s sur/ace, #ic# can not be by c#ance. The relations#i4s s#o ad%ance" (co&4are" to w#at we think t#ey !ne K mathe&atical kno le"ge. Thus, I a& convince" t#ey were t#ere. N#en we analyse t#e layout at Giza, we ha%e to o%erco&e these o(stacles: 1. We are dealing with “as is@, w#ic# is not t#e same as “as reno%ate"” (&ulti4le ti&esO I)JK, #ic# is not t#e sa&e as “as built”, w#ic# is not the same as “as designe".@ 2. As discusse" in a pre%ious paper (....!K, I ha%e co&e to t#e conclusion that Giza was built around 5...! B$E. This is a pro(le&atic date, but it’s w#en t#e star map be#ind t#e design aligns. Figure 1 s#o s ho e6tant Giza aligns with t#e stars around 5...! B$E. Giza ele&ents are in green, w#ile stars an" labels are re", blue and blac!. The “+” sign is t#e celestial nort# pole. 1 Zep Tepi Mathematics 101 I Douglas 2021 >igure 1# Ho( the right hand side o1 Giza aligns with the stars, 5%.%- BC@ This s#o s fi%e pyrami"s, and Thu(an’s or(it around P.. Thu(an was t#e pole star at t#e ti&e. '4art fro& t#e fi%e pyrami"s, I will also s#o ho Cor Caroli, Koc#a(, Du(#e, and P#ec"a 4ositions are mat#e&atically relate" to t#e pyrami"s. They are s#o n as green dots. I don’t kno i/ anyt#ing was built there. Koc#ab in Ursa Minor is actually off the Giza plateau at the mo&ent, (ut it may still have been plateau w#en Giza was built. Koc#ab is close to t#e current entrance to the Ca%e o/ Bir"s. ) Zep Tepi Mathematics 101 I Douglas 2021 S#o ing t#e alignment for si6 pyrami"s is diRicult, because t#e stars invol%e" s4an across more than hal/ the s!y, w#ic# creates pro(le&s trying to map t#e cur%e" s!y to a flat screen or page. The constellations distort differently, de4ending on w#ic# proEection you use. Here are t o atte&4te" alignments using the same s!y and date, just projecte" differently. This is the Mam&er;'itoff projection, with a 101..S fiel" o/ vie , at Cairo on 21 Marc#, 5...! B$E. >igure 2# Giza stellar alignment, HAI projection Mere is the AA$ Zenithal e,ui"istant projection version, wit# the same fiel" o/ vie . Zep Tepi Mathematics 101 I Douglas 2021 >igure !# Giza stellar alignment, ARC projection t# 1. In the years between being built, an" the 4 dynasty, I woul" i&agine that t#e pyra&i"s su=ere" a lot o/ da&age. I inter4ret “K#u/uPs Horizon” as the proEect initiate" by K#u/u, and co&4lete" by K#afre and Menkaure, to “reno%ate and restore” w#at was leD. In t#e process they cleare"t# t#et# area west o/ P2, creating t#e “horizon.” I s4eculate t#at t#ey too! w#at was leD o/ the . and 6 pyrami"s, and possi(ly Dje"e/rePs, and use" t#at to re(uil" t#e other four pyrami"s, s#i44ing in Tura li&estone for the final layer. K#u/u may have had to re(uil" the to4 thir" o/ his 4yrami", fro& t#e area abo%e the granite slabs. Starting fro& there coul" e64lain the graffiti, a twenty year ti&e s4an, and the o"" c#anges in course t#ic!ness. ). Ju"ging by t#e mat#e&atics, MenkaurePs re(uil"ing resulte" in the footprint mo%ing slightly south an" west, and possi(ly re;centring t#e entire pyrami". Legon’s triangle I.J actually ends insi"e t#e pyrami", so I t#ink Menkaure was "ifferently size" to no . .. A/rica has rotate" slightly since 5T..! years ago. This twist, along with MenkaurePs 4ro(le&atic reno%ations, #as co&4licate" any analysis o/ the site. 6. I/ my s4eculation in (1K is correct, t#en having pyrami"s 4, 5 and 6 missing, has also co&4licate" any analysis o/ t#e site. At least we ha%e Nor"en’s docu&entation I1J o/ pyrami" 4 as a starting point. T. Pyrami" 4, 5 and 6 were strongly suggeste" by t#e stellar alignment. A""ing t#e& to the "esign, and w#at co&es out o/ that as docu&ente" here, convinces me that they di" indee" e6ist. Ne can e%en get a goo" i"ea o/ their base sizes fro& the mat#e&atics o/ the site, and potentially their heights.