An Attempt to Create a Proportional Model of “Weave a Home”

Sufian Ahmad

Platonic Solids and Regular Polygons

• Regular polygons are geometric shapes that can be created by dividing a circle into a set of sections

Platonic Solids and Regular Polygons

• A hexagon for example, is created by connecting the intersection points (in red circles) for three circles and the horizon line

Platonic Solids and Regular Polygons

• Basic trigonometry is applicable to regular polygons, they are basically right angled triangles and by applying Pythagorean theorem you can calculate the length of the side (in green)

• Given that we start from a unit circle (r = 1) in the figure to the right the height of the rectangle is: • 12 + X2 = 24

• X = √3 hence giving us the root 3 ratio

Mystic Interpretations of Geometry

• Ancient civilizations believed that geometric properties in platonic solids and polygons are the same values governing the whole universe

• Because these ratios or proportions are very common in nature, they look beautiful to the human eye

• Therefore, it became a traditional “best practice” to follow these proportions in design

Mystic Interpretations of Geometry • Ancients believed that the universe was created the same way we created the hexagon in slide 3, by intersecting circles (the absolute world and the relative world)• That concept of intersecting world is symbolized by the flower of life• Below is another important symbol, Metatron’s Cube• Metatron’s Cube is a hexagon with all the platonic shapes appearing inside it, which symbolizes the creation of the world

Root Ratios

• Root ratios are the relationship between the sides of a proportional rectangle and the sides of another proportional rectangle created inside it

• Lines A and B in figure (green lines)

Root Ratios• To create interesting shapes and ornamentations, geometers

create complex grids by repeating basic proportional divisions over and over

• An example of a golden ratio grid creating an ornamentation

“Weave a Home” Grid

• The basic golden ratio is a simple relationship between two lengths, which is not very beneficial to create an interesting shape

• If we were to only use the basic ratio, we will always have shapes that are divided into two thirds

“Weave a Home” Grid

• To get a close match for the basic “Weave a Home” design (10 bending points on frame) we have to use the full range of ornamentation grids

• By full range of ornamentation grid, I mean instead of just using the golden ratio I will extract ratios from the pentagon and the decagon (10 sides) grid

• A process is similar to traditional ornaments created (slide 8)

Animated Grid• The animation below demonstrates how a golden ratio

ornamentation lines and parts all fit a repeated decagon grid

Polygenic grids and Values• The red lines (pentagon sides)

gradually get smaller and smaller

• The golden ratio governs the relationship between red sides

• Plotting the lengths of the red sides onto a line chart will give us the ratio curve

The Roots Curves

1 2 3 4 5 6 7 80

10

20

30

40

50

60

goldenroot threeroot 2root 3 sidespentagon sides10 sides

Different Roots and polygons

A repeated 12 sided polygon (root 3) A repeated sqaure (root 2) A repeated pentagon (golden)

A repeated 10 sided polygon (golden)A repeated hexagon (root 3)

From Sides to Angles • To divide a right angle into 8

proportional angles rather than proportional lengths I compared the lengths of 8 sides of a repeated polygon grid to 8 angles that sum up to 90

• E.g. A series of Hexagon side lengths with a sum of (visualized in the next slide)

• A simple conversion to sum 90 will give the right column

Lengths Angles

470.094045 25.00313694

352.570534 18.75235272

264.4279 14.06426451

198.320925 10.54819838

148.740694 7.911148802

111.55552 5.933361575

83.66664 4.450021181

62.74998 3.337515886

Sum = 1692.126238 Sum = 90

Hexagon Grid Values

Hexagon Grid Values•The values of 18 sided polygon are plotted as dotted lines vs. 16 sided polygon in solid lines•We have selected the 16 sided polygon convertion