Egyptian temple style: * see Endnote 6

reference to the Egyptian 4th Dynasty, (2613- 2494 BC) , a "golden age" of the Old Kingdom; temples employed absolute straight lines & right angles.

 

sarcophagi relief: * see Endnote 7

Early Christian sarcophagi (stone coffin, especially one bearing sculpture, inscriptions, often displayed as a monument) from Imperial Rome; carvings relating them to early Christianity; produced from the late 3rd to 5th century AD; represent earliest form of large Christian sculpture, are important for the study of Early Christian art.

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Romanesque construction & ornament: * see Endnote 8

(6th -11th century AD) architectural style of medieval Europe; despite regional characteristics & different materials, it is the first pan-European architectural style since Rome; known for its massive quality & signature semi-circle arches, thick walls, sturdy pillars, groin vaults, large towers and decorative arcading; uses clearly defined forms, regular, symmetrical plan; overall appearance is one of simplicity.

 

Gothic cathedrals (mathematics in stone): 

reflects geometry & proportion in stone, from labyrinths in mosaic floor tiles to the crisscrossing ribs that hold the ceiling; the building itself is the geometry text, written by the master mason, with his fingers touching stone, using a simple pair of dividers and carpenters square.  The balance of mass & space goes by square roots of 2 and 3, and the so-called Golden Section. 

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[source:  Mathematical Models of Gothic Structures, by Javier Barrallo & Santiago Sanchez-Beitia]

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Doric temples (mathematics in stone): * see Endnote 9 

The most famous Doric temple ever built is the Parthenon on the acropolis of Athens.  The architects were Ictinus & Callicrates, the project manager was the sculptor Phidias.  It was completed in 438 BC although decoration of the building continued until 432 BC.  Starting with the Pythagoreans, the Greeks attributed great meaning to certain Platonic solids (the regular polyhedrons).  The attached great meaning to the Golden Section or Golden Ratio.  This ratio is can be expressed geometrically; we express this as the irrational number Pi.  For the Greeks it reflected philosophy & wisdom.  Since the Parthenon was dedicated to Athena, the Goddess of Wisdom.  We find the Golden Ratio repeatedly used in the dimensions of the Parthenon.

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canon of the statue:* see Endnote 10

Greek sculptor Polykleitos (c.450–420 BC) famous for his ideally proportioned bronze Doryphoros, wrote the Canon describing proportions to be followed in sculpture, applies the mathematical concepts of geometry (ratio, proportion & symmetry) to describe the human form. Lysippus (c.395-305 BC) modified these; his figures use slimmer proportions, he reduced size of the head & elongated the limbs, his figures appear taller, more statuesque.

 

Doric order:

strictly governed by established geometric proportions (not the architect’s aesthetic taste); a façade might be 2, 4,6, 8 or 10 columns wide but the relationship between the front & side columns followed the ratio twice the number of frontal columns +1.  The diameter of the base of a column was a standard unit of measurement.  The height of the entablature was 2 diameters.  The height of columns was 8 diameters.  The space between columns was based on the diameter of the column.

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space representation:

see glossary page 7, Introduction

 

pyramid temples, dimensions: * see Endnote 11

despite its imposing size, mass and the building project’s long duration, the Great Pyramid at Giza has amazing fidelity in terms of accuracy & proportion; this accuracy was part of the built in design though the written evidence for this (Egyptian or secondary Greek, Roman) is missing.

 

Egyptian building techniques:  

the most challenging building the Egyptians engaged in was the construction of the Great Pyramids; there are no absolute conclusions as to how this was done through the use of levers and ramps seems almost certain.  Ancient sources (Greek 5th century) suggest pyramids were constructed like stairs or tiers; when a level was completed workmen used short wooden logs as levers to raise the rest of the stones (portable levers of wood) lifting the blocks from the ground onto the first tier; these stones resting atop the first tier were used for the next level and when this was complete the process was repeated; the finishing process was done from the top down.

 

Egyptian water control:

basin irrigation water management system, controlled rise & fall of the Nile to suit agricultural needs; a crisscross network of earthen walls formed in fields which would be flooded by the Nile.  Flood water was trapped in the basins; they would hold water longer than it would have naturally stayed, allowing soil to become saturated for later planting; once saturated remaining floodwater was drained to another basin needing more water.

 

Egyptian administration: 

see glossary page 7, Endnote 8 Introduction.

 

Egyptian calendar:

solar calendar, 365-day year, 3 seasons, 120 days each, each season had 4 months 30 days; months came to be known by the names of their festivals; established early in Old Kingdom, evidence of its use early in the 4th Dynasty (reign of Shepseskaf, c. 2510 BC); based upon astronomical observations of Sirius (whose reappearance in the sky closely corresponded to the average onset of the Nile flood through the 5th and 4th millennium BC).

 

Ahmes:

(1600- 1500 BC) Egyptian scribe, during 15th Dynasty (2nd Intermediate Period) and 18th Dynasty (and of the New Kingdom); wrote the Rhind Mathematical Papyrus, on Egyptian mathematics, 1550 BC; the earliest contributor to mathematics whose name is known.