## FIRE ALTARS OF THE VEDAS – A LOGICAL PUZZLE FROM ANCIENT INDIA

“3D Thinking” Pages 67 to 72: Vedic Fire Altars present a logical puzzle for those building them. Firstly, the surface area has to be in direct proportion to the height of the individual whose task it is to construct the altar. Secondly, there is to be a precise alignment of the altar with the fourContinue reading FIRE ALTARS OF THE VEDAS – A LOGICAL PUZZLE FROM ANCIENT INDIA

## THE PYTHAGOREANS: PYTHAGOREAN TRIANGLES & CIRCLES

“3D Thinking” Page 131: An animation of the Pythagorean Theorem shown dynamically in terms of circles or spheres. In the animation the diameter of the outer circle/sphere equals the hypotenuse of the right-angled triangle. The diameters of the two inner circles/spheres equal the lengths of the other two sides of the right-angled triangle. According toContinue reading THE PYTHAGOREANS: PYTHAGOREAN TRIANGLES & CIRCLES

## LABRYNTHS MYSTERIES and METHODS

“3D Thinking” Page 32 Neolithic Geometries: Labyrinths are constructed with a single path that winds forwards and backwards, multiple times, before arriving at a central point. The oldest surviving labyrinths appear as Neolithic and Bronze Age rock carvings – a petroglyph labyrinth in Usgalimol, India has been estimated to be 10,000 years old. By the time of the middle ages labyrinthsContinue reading LABRYNTHS MYSTERIES and METHODS

## Geometry Ancient Egypt – Pyramid Proportions

“3D Thinking” Page 74 to 87: The slope ratios, dimensions, construction, and alignment methods, of the Pyramids of Ancient Egypt: THE SLOPES OF THE PYRAMIDS OF EGYPT Given the whole number traditions of ancient Egypt one would expect the slopes of the pyramids to be in whole number proportions too. Whole number grids of squareContinue reading Geometry Ancient Egypt – Pyramid Proportions

## A Geometry of Early Islam – Nesting Polygons

“3D Thinking” Page 190, 191: There are a number of Islamic design methodologies of which the “Nesting Polygon” method is one. The method places an arrangement of polygons within a polygon and continues infinitely placing the same but scaled arrangement within each polygon as it appears. The method was used to generate a lattice fromContinue reading A Geometry of Early Islam – Nesting Polygons

## A Geometry of Early Islam – ABJAD, Circles and a Door

“3D Thinking” Pages 175 to 177 Islamic Geometry: This Seljuk period door dates from 13th-century Anatolia and stands as a possible example of a use of numeric values to communicate a message. The door’s design is based the Islamic “close-packing circle” method of design – in this case an arrangement of close-packing circles within a pentagon. The two primary numbersContinue reading A Geometry of Early Islam – ABJAD, Circles and a Door

## A Geometry of Early Islam – The Dynamic Circle Method

“3D Thinking” Pages 187, 274: Using close-packing circles to create surface designs, during the early years of Islam, requires a method to generate different close-packing circle arrangements. The dynamic sphere geometry provides such a method. Applying the geometry, starting with a 5-circle arrangement within the unit triangle of a square, generates many arrangements including theContinue reading A Geometry of Early Islam – The Dynamic Circle Method

## A Geometry of Early Islam – The Circle Method and a Window

“3D Thinking” Page 187: The animation below shows the Arabian “close-packing” circle method that was used to create the window design of the 1356CE Madrasa of Amir Salf al-din Sargatmish in Old Cairo, Egypt. The design method dates back to the Abbasid period – a time of great creativity. The development starts with (i) aContinue reading A Geometry of Early Islam – The Circle Method and a Window

## Geometry Divine Proportions

“3D Thinking” Pages 67, 75, 77, 210 to 213: Ideas of “Divine Proportions” originated in the distant past. For the Pythagoreans whole numbers resonated throughout the cosmos in sacred music, in geometry and in number. For the Harrapans of the Indus Valley they found expression in fire altars of different shapes but of constant areasContinue reading Geometry Divine Proportions

## A Geometry of Early Islam – The “Ray” Method

“3D Thinking” Pages 183 to 186: An example of the Ray Method – one of the six primary methods used to create Arabian designs during the early Islamic period – during the Abbasid Caliphate 566-653CE. In the animation one arm of a “10-Ray” is used to position a “6-Ray” – then on the intersection ofContinue reading A Geometry of Early Islam – The “Ray” Method