Got To Love Geometry!

Got to love Geometry!

The Complex Geometry of Islamic Design

In Islamic culture, geometry is everywhere. You can find it in mosques, madrasas, palaces and private homes. This tradition began in the 8th century CE during the early history of Islam, when craftsman took preexisting motifs from Roman and Persian cultures and developed them into new forms of visual expression.

This period of history was a golden age of Islamic culture, during which many achievements of previous civilizations were preserved and further developed, resulting in fundamental advancements in scientific study and mathematics. Accompanying this was an increasingly sophisticated use of abstraction and complex geometry in Islamic art, from intricate floral motifs adorning carpets and textiles, to patterns of tile work that seemed to repeat infinitely, inspiring wonder and contemplation of eternal order.

Despite the remarkable complexity of these designs, they can be created with just a compass to draw circles and a ruler to make lines within them, and from these simple tools emerges a kaleidoscope multiplicity of patterns. So how does that work? Well, everything starts with a circle. The first major decision is how will you divide it up? Most patterns split the circle into four, five or six equal sections. And each division gives rise to distinctive patterns.

There’s an easy way to determine whether any pattern is based on fourfold, fivefold, or sixfold symmetry. Most contain stars surrounded by petal shapes. Counting the number of rays on a starburst, or the number of petals around it, tells us what category the pattern falls into. A star with six rays, or surrounded by six petals, belongs in the sixfold category. One with eight petals is part of the fourfold category, and so on.

There’s another secret ingredient in these designs: an underlying grid. Invisible, but essential to every pattern, the grid helps determine the scale of the composition before work begins, keeps the pattern accurate, and facilitates the invention of incredible new patterns. Let’s look at an example of how these elements come together.

We’ll start with a circle within a square, and divide it into eight equal parts. We can then draw a pair of criss-crossing lines and overlay them with another two. These lines are called construction lines, and by choosing a set of their segments, we’ll form the basis of our repeating pattern.

Many different designs are possible from the same construction lines just by picking different segments. And the full pattern finally emerges when we create a grid with many repetitions of this one tile in a process called tessellation.

By choosing a different set of construction lines, we might have created this any of the above patterns. The possibilities are virtually endless.

We can follow the same steps to create sixfold patterns by drawing construction lines over a circle divided into six parts, and then tessellating it, we can make something like the above.

Here’s another sixfold pattern that has appeared across the centuries and all over the Islamic world, including Marrakesh, Agra, Konya and the Alhambra.

Fourfold patterns fit in a square grid, and sixfold patterns in a hexagonal grid.

Fivefold patterns, however, are more challenging to tessellate because pentagons don’t neatly fill a surface, so instead of just creating a pattern in a pentagon, other shapes have to be added to make something that is repeatable, resulting in patterns that may seem confoundingly complex, but are still relatively simple to create.

This more than 1,000-year-old tradition has wielded basic geometry to produce works that are intricate, decorative and pleasing to the eye. And these craftsman prove just how much is possible with some artistic intuition, creativity, dedication along with a great compass and ruler.

Geometry apps!

I have stumble upon some of the neatest apps, for a person like me that love geometry, these apps are just what I’ve been looking for!

Euclidea:

IOS: https://itunes.apple.com/app/id927914361 Android: https://play.google.com/store/apps/details?id=com.hil_hk.euclidea This app is only Euclidean geometry, that’s what I like the most. Appinfo: Euclidea is a brilliantly original way to learn about, explore and have fun with Euclidian Constructions! Your task is to solve interesting challenges by building geometric constructions with a straightedge and compass. If you design the most elegantly simple solutions in the least number of moves, you’ll earn the highest scores. Solutions are scored in lines (L) and elementary Euclidean constructions (E).

XSection:

IOS: https://itunes.apple.com/us/app/xsection/id1069933287?mt=8 Haven’t found it on Android yet, but this app aims to learn you about Polyhedrons, cross sections and geometry. Appinfo: Learn how to construct cross sections of polyhedrons. Study different techniques, train yourself, and then solve geometric puzzles.

Pythagorea:

IOS: https://itunes.apple.com/us/app/pythagorea-geometry-on-square/id994864779?mt=8 Android: https://play.google.com/store/apps/details?id=com.hil_hk.pythagorea&hl=sv Just found this game, so I paste the appinfo: Pythagorea is a collection of geometric puzzles of different kind that can be solved without complex constructions or calculations. All objects are drawn on a grid whose cells are squares. A lot of levels can be solved using just your geometric intuition or by finding natural laws, regularity, and symmetry.

Pythagorea60:

IOS: https://itunes.apple.com/us/app/pythagorea-60-geometry-on/id1043064990?mt=8 Android: https://play.google.com/store/apps/details?id=com.hil_hk.pythagorea60&hl=sv Just found this game, so I paste the appinfo: Pythagorea 60° is a collection of more than 270 geometric problems of different kind that can be solved without complex constructions or calculations. All objects are drawn on a grid whose cells are equilateral triangles. A lot of levels can be solved using just your geometric intuition or by finding natural laws, regularity, and symmetry.

Horis International Limited

All these games are made from the same developer, Horis International Limited: IOS: https://itunes.apple.com/us/developer/horis-international-limited/id646324304 Android: https://play.google.com/store/apps/developer?id=HORIS+INTERNATIONAL+LIMITED Big up, love this!

Double Arm

This sketch was made by rotating point 1 around a circle, rotating point 2 around point 1, and then joining a line between point 1 and point 2. In the gif point 2 is traveling in the same direction as point 1. Point 1 is traveling at 1x, and point 2 is traveling at 2x. Following the gif are some of my favorite combinations of the periods:

Rotating same direction:

1x and 1x

1x and 2x

1x and 5x

2x and 1x

2x and 5x

5x and 8x

The last image is an array of all the varieties from 1x to 9x for each point.

code: http://p5js.sketchpad.cc/sp/pad/view/jLceHatXid/latest

23 notes View Post

mrvmt

9 years ago

Here’s the orginal creator, he’s a legend:

beesandbombs (http://beesandbombs.tumblr.com/)

Take care

Waving Circles

Remake of this awesome gif (does anyone know the original creator?): https://33.media.tumblr.com/a5d2523e8a4fc92e1629ae9937d870f6/tumblr_nl4lm8O73g1u93xcqo1_500.gif

This is early days for my conversion to p5.js. Here’s the code: http://jsfiddle.net/dandersod/8uL4s35d/

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beesandbombs

81 notes View Post

mrvmt

7 years ago

Wow, This was really intresting

929 notes View Post

mrvmt

7 years ago

Glad to see your back. Love your work. As always, keep up the good job! And btw, nice covergirl photo ;)

Rough Draft Of A Sculpture Illustrating The Electric Field Lines Between Two Equal And Opposite Charges.

Rough draft of a sculpture illustrating the electric field lines between two equal and opposite charges.

(Hi! I’m still alive, sorry for ghostin ya tumblr. I stopped posting because my reclusive tendencies have gotten really out of control over the past year or so, but I’m trying to take little baby steps to quittin my bullshit so bear with me! Still makin math art like crazy, so I thought I’d start with sendin u this lil ditty I whipped up a couple weeks ago. Plenty more where this came from, I promise!)

181 notes View Post

mrvmt

8 years ago