A while back there were some posts about generating Chladni Figures using R scripts. These scripts generated some pretty nice images, I thought. But, to experiment a bit more it would be nice to have something interactive, so I put together this page, which you can use to make images like the one below.

Adding more vibrations to the surface, you can get some pretty intricate looking patterns:

If you would like to try it out, please visit: https://dmackinnon1.github.io/chladni/ (source here).

# mathrecreation

## Monday, July 17, 2017

## Saturday, April 29, 2017

### truchet en plus

Since the previous post, I have been playing around with more variations on Truchet tiles (using this page). The variety of appealing patterns that you can create from these simple tiles is impressive.

For example, using this simple base tile you can create a path-like effect, even to the extent that paths can seem to weave under and over each other. The patterns below use this effect to suggest links and knots.

Slight variations in the base tile can produce interesting effects. Here's an example that uses the traditional Truchet base tile.

Bulging the dark right triangle into a quarter circle allows us create something that looks quite different, even though it consists of exactly the same tile placements. Squares are replaced by circles, V-patterns replaced by tulips, and we end up with organic and densely packed patterns.

A common variation on the Truchet tile is the Smith tile, which consists of two quarter circles at opposite corners. Unlike the traditional Truchet, the Smith base tile has 180 degree symmetry, but because it lacks 90 degree symmetry, it can still be used to produce interesting and appealing patterns (if we had quarter circles in all four corners, the constructed patterns would always be the same - an array of circles). Using this tile, we end up with circles and blobby regions that create paths and zones across the grid.

A small change to the Smith tile allows us to eliminate the 180 degree symmetry and regain the expressiveness of the traditional tile patterns. For example replacing one of the quarter circles with a square means that some circles in our original pattern become squares, some remain circles, and some take on a lemon-shape, while the overall pattern retains the same topology as it has in the Smith version.

Another small change (using a diagonal line in place of the square) produces patterns that look quite different at fist: our paths now resemble strange jigsaw shapes. A closer look shows how the essential features are retained.

Even with all the possible variations, the original tile retains its charm. Can you see the four trefoils in the pattern below?

*the humble Truchet tile*

For example, using this simple base tile you can create a path-like effect, even to the extent that paths can seem to weave under and over each other. The patterns below use this effect to suggest links and knots.

*Truchet patters for two links (left)*

and a trefoil knot (right)

and a trefoil knot (right)

Slight variations in the base tile can produce interesting effects. Here's an example that uses the traditional Truchet base tile.

A common variation on the Truchet tile is the Smith tile, which consists of two quarter circles at opposite corners. Unlike the traditional Truchet, the Smith base tile has 180 degree symmetry, but because it lacks 90 degree symmetry, it can still be used to produce interesting and appealing patterns (if we had quarter circles in all four corners, the constructed patterns would always be the same - an array of circles). Using this tile, we end up with circles and blobby regions that create paths and zones across the grid.

A small change to the Smith tile allows us to eliminate the 180 degree symmetry and regain the expressiveness of the traditional tile patterns. For example replacing one of the quarter circles with a square means that some circles in our original pattern become squares, some remain circles, and some take on a lemon-shape, while the overall pattern retains the same topology as it has in the Smith version.

Another small change (using a diagonal line in place of the square) produces patterns that look quite different at fist: our paths now resemble strange jigsaw shapes. A closer look shows how the essential features are retained.

Even with all the possible variations, the original tile retains its charm. Can you see the four trefoils in the pattern below?

## Sunday, March 26, 2017

### truchet tiles

A short while back, I posted about the images found in books about Froebelian kindergarten exercises, like

A surprising variety of patterns can be formed by restricting things even further, considering the case where each square is cut in half along a diagonal with half of the square coloured black, the other coloured white (i.e. no blank or completely filled squares, as are found in the images above).

Arrangements of these tiles were studied extensively by Sebastien Truchet, whose book on the subject (

Truchet tiles, as they became known, have been studied extensively and generalised to include other tile sets that are not rotationally symmetrical. In the image below, we start with a traditional Truchet tiling, then only show the diagonals, and finally replacing the diagonals with quarter circles, centred around the vertices where the diagonals used to touch (these tiles, introduced by Cyril Stanley Smith, create interesting patterns of blob-like paths and circles).

To play around with these I've put together a "Truchet Tiles" page here.

There are a number of illustrations in Truchet's text that are worth checking out, and you can reproduce them using the page mentioned above. Here is tiling "38" from Truchet:

Here is the same tiling using the Truchet tile page, also rendered using only diagonals and the Smith tiles.

There are lots of questions that can be asked about arrangements of the tiles. Truchet enumerates some of the possible arrangements using symbols and illustrations - below are the first two tables of rows of tiles that he lists (how many will be in the next two tables?).

Here is another rendering of one of Truchet's original patterns, number 52:

And replacing the traditional truchet tiles with similarly oriented diagonal and Smith tiles - as you might have noticed, doing this looses information by a factor of 2 for each tile:

Try out the Truchet tile page here: https://dmackinnon1.github.io/truchet/.

**The Paradise of Childhood**(on Google Books here). These old books provide great examples of patterns and designs that can be easily drawn by hand with graph paper, in many cases only using arrangements of congruent 45-90 triangles.*Nineteenth Century Froebelian Doodles*

A surprising variety of patterns can be formed by restricting things even further, considering the case where each square is cut in half along a diagonal with half of the square coloured black, the other coloured white (i.e. no blank or completely filled squares, as are found in the images above).

*Some arrangements of four Truchet Tiles*

Arrangements of these tiles were studied extensively by Sebastien Truchet, whose book on the subject (

*Method for creating an infinite number of different designs with squares halved into two colours along a diagonal*) can be found online here.Truchet tiles, as they became known, have been studied extensively and generalised to include other tile sets that are not rotationally symmetrical. In the image below, we start with a traditional Truchet tiling, then only show the diagonals, and finally replacing the diagonals with quarter circles, centred around the vertices where the diagonals used to touch (these tiles, introduced by Cyril Stanley Smith, create interesting patterns of blob-like paths and circles).

*Three popular variations on Truchet tiles:*

traditional, diagonal, and semi-circles

traditional, diagonal, and semi-circles

To play around with these I've put together a "Truchet Tiles" page here.

There are a number of illustrations in Truchet's text that are worth checking out, and you can reproduce them using the page mentioned above. Here is tiling "38" from Truchet:

*tile pattern 38 from Truchet*

Here is the same tiling using the Truchet tile page, also rendered using only diagonals and the Smith tiles.

*tile pattern 38, generated here*

There are lots of questions that can be asked about arrangements of the tiles. Truchet enumerates some of the possible arrangements using symbols and illustrations - below are the first two tables of rows of tiles that he lists (how many will be in the next two tables?).

And replacing the traditional truchet tiles with similarly oriented diagonal and Smith tiles - as you might have noticed, doing this looses information by a factor of 2 for each tile:

Try out the Truchet tile page here: https://dmackinnon1.github.io/truchet/.

*Update: More truchet fun in this post.*## Friday, March 24, 2017

### probability simulations using R

An ongoing side project of mine is learning how to use the statistics scripting language R, and have been putting putting together R markdown files that set up simulations for a variety of probability problems. You can find some of them here: https://dmackinnon1.github.io/r_examples/.

These are simulations that generate data based on problems like the Birthday problem, the Monty Hall problem, and the Burnt Pancake problem. Learning scripting languages aside, it is always good to keep digging into probability problems: they confuse practically everyone.

These are simulations that generate data based on problems like the Birthday problem, the Monty Hall problem, and the Burnt Pancake problem. Learning scripting languages aside, it is always good to keep digging into probability problems: they confuse practically everyone.

## Tuesday, March 7, 2017

### Ulam's two step cellular automata

Above are some of the nice images generated by a cellular automata described in one of Martin Gardner's essays about Conway's Game of Life (you can find the essays here). Cells have four neighbours (north, south, east, west), and follow only two rules that are applied at each step: if a cell has one live neighbour it turns on, and if a cell is on it turns off after two steps. The images above start happening around step 100 after turning on a single cell at the centre of a 61 by 61 grid.

You can play with these here. Eventually, these will start to repeat or disappear completely (I suspect they will oscillate, but have not found out when yet). On a 5 by 5 board, a single central cell will lead to a pattern that dies out in 10 generations; once you get to the uniform checkerboard state, the next is an empty board.

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