Tesselations: Squares, hexagons and implied dodecagons.

Tesselations: Dodecagons, squares and implied hexagons

Tesselations: Dodecagons, hexagons and implied squares.

This week's tesselation uses the dodecagon, the forgotten polygon in the tesselation universe, and the hexagon. The empty spaces are squares.

Doing a little color adjustment with a second photo using flash, this is what it might look like at night.

I wish I had enough of these to cover a large wall with them. I like these a lot.

Tesselations: Hexagons, squares and triangles at maximum contrast

Two quick ideas for this week. I made the patterns on a table that was too busy, so I pumped the contrast to turn it black.

There's more to come with the hexagons squares and triangles.

Tesselations: A new pattern of hexagons, squares and triangles

Hexagons, square and triangles can make another more regular pattern, but this one works as well. The difference between this one and the other can be seen when you look at the hexagons. Yellow hexagons have a green square on each edge, while the green hexagons are surrounded by alternating triangles and squares.

I also made this small example using three colors, but the intricacy of the pattern make adding more colors a little problematic. I will return to this when I find a good color scheme with more than two colors.

Tesselations: Variations of lineoleum

Ocatgons and squares are a standard tiling pattern using linoleum. I'm sure everyone has seen these many times.

Often, the octagons are all one color and the squares are another single color.

In this version, there are two colors of octagons and the same two colors of squares.

Now, two colors of octagons and a third color for all the squares.

And finally, three colors of octagons and a fourth color for all the squares.

Triangles and hexagons. Take two.

Using yellow hexagons, green, blue and yellow triangles, very little color manipulation.

Blue hexagons, green and yellow triangles, lots of color manipulation.

Both triangles and hexagons in green and black only.

I like this one. I shrunk it down with the idea it could be a logo. There are only two sizes of triangles here, but the large aquamarine have a little bit of an optical illusion going on, with some looking smaller and others larger.

Hexagons, squares and triangles. More variations on the theme.

More color experiments with the tiling pattern that uses hexagons, squares and triangles.

This is the simplest of the three variations and some may find it the most satisfying.

Another variation that uses only black hexagons, but more colors of squares and triangles.

And a third pattern where black tiles are banned, working just with the yellow, green and blue.

Yellow squares, black triangles and hexagons.

Here is yet another tiling pattern that I think should be better known than it is, the triangle, square and hexagon pattern, this time done in just two colors. The buffer zone around each hexagon, created by the squares and triangles, forms a regular dodecagon, a twelve sided polygon where every corner angle measures 150°.

Isn't it pretty?

Dodecagons, squares and triangles.

This is analogous to using octagons and square, the famous linoleum tile pattern, but dodecagons when lined up north/south/east/west leave a gap filled by triangles and squares, not just a simple square.

The same idea with the squares removed, which makes it a little hard to line the triangles up neatly. Without magnetized backs, these shapes can be very delicate to work with, and lining things up perfectly becomes more of a challenge.

Dodecagons, hexagons and squares.

The big blue and green shapes have twelve sides each, so they are dodecagons. The yellow hexagons and black squares should be more familiar.

I really like tesselations, as should be obvious to any regular reader by now.

Here's a close-up where none of the wooden table is visible.

Triangles and hexagons. Simple patterns that should be better known.

Because every angle of an equilateral triangle is 60° and every regular heaxgon has angles of 120 °, the two shapes mix and match in several ways. If I use less colors than I did here, the Star of David motif in this pattern becomes much more prominent.

Here is a completely different pattern where a buffer of triangles surrounds each hexagon. This is my favorite of the regular mix and match patterns using triangles and hexagons, and I'm going to do more experimenting with color options with this one over the next few weeks.

Two versions of Sierpinski's Gasket, using the new toys.

The standard version of Sierpinski's gasket is two colors. It is a simple fractal that is supposed to be created by infinite recursion, but I only have finitely many triangles, so this will have to do.

Here is a four color version of the gasket, using Pascal's triangle and mod 4 arithmetic. Four colors lets me make a bigger version, but some color will eventually be in short supply, this time it was the aquamarine.

I have an idea for how to make this using empty space as the most used shape. That should make a bigger version.

A little message to my friends and family at the Fourth of July party.

Using my new toys like an old 8-bit programmer to send a shout out to all my peeps.

Thanks for all the great food and wonderful companionship. I had a great time. See you on Labor Day if not sooner.

Now officially in the "I'm never gonna get tired of these!" stage with my new toys.

Okay, I'm obsessed with the new toys. Like Princess Sparkle Pony loves Ernie Bushmiller and Ugly Tour Buses, like Padre Mickey loves Red Mr. Peanut Bank and The World's Most Beautiful Grandchile™, I'm in love with the new toys from SeriousPuzzles.com.

The polygons that come with the set have 12 sides, 8 sides, 6 sides, 4 sides and 3 sides, but there is a way to use octagons and triangles to make a negative space regular 24-gon. The little white added rhombi are 30°, which fits the outside gaps perfectly and makes it easier to build. The shape looks like a flower, obviously, but the influence of The Other Blog is making me think of some fabulous necklace Miss Elizabeth Taylor might have received from a suitor.

In any case, I'm going to try this shape again with different colors and different backgrounds. Artists call this "variation on a theme". My readers may soon call it "beating a dead horse".

I am going to invoke the First Rule of Blogging, which of course is "You're not the boss of me."

Enjoy.

The new toys have arrived, Part 2.

Yay, even more new toys have arrived. These are regular polygons that can be used to fill an infinite plane with a repeating pattern.

These shapes are not magnetic like the Penrose tiles, but they come in four colors and five shapes, regular polygons with three sides (equilateral triangles), four sides (squares), six sides (regular hexagons), eight sides (regular octagons) and twelve sides (regular dodecagons).

A few years back, I drew pictures of some of the possible mix and match patterns that create regular tilings of the plane, and now I can re-create these with the new toys. This shot shows that the number of pieces used is finite.

But if I take a close-up, you can see that this could be part of a pattern that repeats indefinitely.

Thanks again to the folks at SeriousPuzzles.com, where I bought my new toys.

New toys! So much fun.