2-D Tilings on the Square Grid

Using only the square grid, straight lines that run along its axes of symmetry and quarters of circles contained in its tiles, you can create infinitely many tilings. They can use a single set of tiles, or several different ones like in the example on the left. But if it is periodical, what is the 'fundamental region', the smallest area you need to reproduce the entire tiling?

2-D Tilings on the Triangular Grid

Similarly, using the regular triangular grid as a template for designing tilings, the possibilities are mesmerizing, though the rules are different.

Starting with the Mitsubishi logo (it means three diamonds in Japanese!), what happens when you try to tile it?

2-D Tilings: the Islamic Tradition

... And then there is the Islamic tradition. The floor and wall tilings of the Islamic world demonstrate a very intricate and complex knowledge of the symmetries of two-dimensional space. Their use of regular and irregular, convex, concave and stellated polygons is spectacular!

3-D Tilings on the Cubic Grid

Once your system of generating rules is set in 2-D, you can translate it into 3-D and obtain three-dimensional tilings that work essentially the same way. In the example on the left, each 'tile' is made with sections of sphere, concave and convex (so that it interlocks with itself!) and some sections of planes.

3-D Tilings on the Tetrahedral-Octahedral Grid

Cubes fill space.
Attach two regular tetrahedra to the opposite faces of a regular octahedron, and you obtain a parallelipiped which in turn fills space, so regular tetrahedra and octahedra fill space together. Use this as a basic grid, and you can create endless combinations of 3-D space-fillers.

3-D Tilings: Islamic Ceilings

The Islamic artisans did something similar to the 3-D Tilings on the Cubic Grid with their tilings: the ceilings of many buildings show extrusions of some of their tilings. With phenomenal results!

The Geometry of Paper-Folding

Paper does not stretch to any significant extent. This means that it does not act as a topological object, but as an affine one. So what are the rules? In Origami, the shape of the boundary will have an effect on the geometry since it gives the folder reference points like middle, half-point, bisector, and so on.

Square Modular Origami

Classical square modular Origami is a great way to learn about polyhedra. Here is an example of its use for a look at a diagonal tennis-ball curve on a cube.

Single Unit Telescopic Square Origami: The Temple

Multiple Unit Telescopic Square Origami: the Lotus

I used seven squares of paper of progressively smaller size to make this lotus flower. The area of each piece of paper was ½ the area of the previous one. Without using a measuring device, the diagonal line of each smaller piece of paper was the length of the edge of the previous one. All the pieces are folded the same way and inserted into each other.

Band Origami

Bands of paper such as rolls from cash registers can be folded into all manner of shapes. They are particularly exciting for modular and periodic foldings.

Circular Origami

In Origami, the shape of the perimeter determines the possibilities for reliable folding, and with circles, it helps that arcsin ½ = 30°

3-D Hilbert's Space-Filling Curve in Origami?!

That's right: how to build an Origami space-filling curve model. Ref.: Mathematics and the imagination/Edward Kasner and James Newman

Knots and Interlaces

Interlaces are drawings of complex knots. In general, each strand follows an over-under-over-under path inside a restricted area. In the example on the left, there are two 'strands' crossing each other in sixteen places. Knots are also the subject of a mathematical domain called Knot Theory.

More Fun with Space-filling Curves

Recently, I began combining interlace designing with the space-filling curves of mathematical fame. The example on the left shows a superposition of two instances of a level 3 Hilbert curve. The two instances are rotated and translated relative to each other, then the two strands are interwoven by an over-under sequence. The negative spaces were then colred according to their size and positions relative to the whole. The result was animated.

Mazes and Labyrinths

Besides challenging the differentiation between linearity and non-linearity, mazes and labyrinths are often based on hidden geometric structures and simple algorithms. I 'discovered' the algorithm illustrated here many years ago by chance and have since tried it on the regular and deformed triangular and square grid, beginning with boundaries that are shaped like triangles, squares, rectangles, parallelogrammes and hexagons.

Calendars and Representations of Time

The astral bodies that we use to measure time are not synchronized. The path of the earth around the sun takes 365.242190 days. In 1994-2000, the moon took 27.321582 days to complete its cycle. The relationship between the cycles of movement in space is irrational. Things never quite fit together. Emphasizing the different cycles or their relationship allows you to create many different calendars.

Project Geraldine: Barn Raising
an Endo-Pentakis-Icosi-Dodecahedron

The large scale of the barn-raising was designed expressly to give the participant a new point of view with regards to polyhedra. Instead of the "God's eye view" of hand held shapes, or the tunnel vision allowed by computer modelling, the barn-raising gave the opportunity to observe the deltahedron on a human scale, where the shape's structure is tangible in terms of the whole body, not limited to the finger tips and the eye.

Twirlahedra: Designing a Construction Kit for Deltahedral Families

This kit for constructing deltahedra uses the first eight subdivisions of the equilateral triangle into smaller equilateral triangles. It allows the building of families of deltahedra having some of the same symmetries.

Creative Writing

As Natalie Goldberg says: "Writing is the crack through which you can crawl into a bigger world..."

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Glass

Glass can be both hard and soft, clear or opaque, smooth or jagged, colored or plain textured or flat. It is a wonderful medium becasue it can be worked in so many different ways: it can be molded, carved, blown or even etched.

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