Since deciding to renew our lease for a second year, we’ve been trying to improve our apartment in several small ways: getting fabric for curtains, moving the recycling from a paper bag in the coat closet to a bin under the sink, re-gluing sections of doors that have warped apart, etc. One project we’ve had in mind for years is to set up a magnetic spice rack, and we finally did it: we attached magnets to the lids of small mason jars, popped our spices in, and stuck the jars onto the side of our refrigerator.
We did feel a pang of nostalgia at putting the Dutch jars with labels like tijm and koriander into the new recycling bin. (That would have been even harder if we’d brought any dragon back with us.) But the result is beautiful!
But as soon as the question of how to arrange the jars occurred to us, Clara and I were both reminded of a game we’ve been playing called Blendoku. The idea of Blendoku is that you’re presented with several blocks of color, and you have to put each one in the right place so that the colors gradually shift from one extreme to another:
In a single row or column, the colors are just a straightforward progression from one extreme to the other. But the more complicated Blendoku levels require you to arrange the colors to fit along multiple gradations at once, like a crossword puzzle. This 30-second trailer gives you a pretty good idea of the gameplay:
My favorite type of Blendoku is one where the colors are all to be arranged in one big grid, like this:
Each row and column is a straight color progression from one end to the other. But instead of using a square grid like Blendoku uses, we are trying to arrange our spices in a hexagonal grid, where the jars form three sets of lines, one horizontal and two diagonal, and it would make sense to ask for the jars along all three sets of lines to be straight progressions. What kinds of colors families would fit in such a three-way crossword?
The answer involves a closer look at the possibilities for Blendoku grids, and a little bit of differential geometry!
Let’s first make the notion of “straight color progression” a little more precise. Humans see three dimensions of colors, according to how much a color activates the red, green, and blue cones in our eyes, so a given color can be plotted as a point in a three-dimensional space:
I took screenshots of some Blendoku puzzles I’d solved, used a color picker to pull out the amounts of red, green, and blue in each of the colors, and then used a 3d-plotter to show where all the colors were in color space. Here are plots of the colors in the “straight line” and “grid” puzzles:
The mustardy top color is mostly green and red, the bottom mostly green and blue, and the dots in between form a pretty straight line from one to the other.
These sixteen colors aren’t random: as points in red-green-blue colorspace they neatly form a 4×4 grid all by themselves.
But some grids don’t look like neat arrays in colorspace:
I had to add lines connecting the dots, and look at it from different angles, to see what was going on.
Along each row or column of the grid, the dots do form roughly straight lines, but the surface they make as a whole is curved like a saddle, or a Pringles chip. Here’s a similar surface where you can see more clearly that each horizontal and vertical strip is straight:
A surface made up of a family of straight lines is called ruled, and if it can be ruled in two different ways it’s called doubly ruled. If we plotted the surface formed by a hexagonal Blendoku grid, it would have to be triply ruled, since each set of rows or diagonals would give a family of straight lines on the surface.
But differential geometry tells us that there is no triply ruled surface! Or rather, that every triply ruled surface is a plane. (The reason: besides planes there are actually very few doubly ruled surfaces to start with—just “one-sheet hyperboloids” and the “hyperbolic paraboloid” shown above—and none of those can be ruled in a third way.) Therefore all the colors in a hexagonal Blendoku grid must lie on a plane in colorspace.
Well, we did our best. Here’s what we came up with:
Unintentionally, spices we to tend use together ended up near each other: curry spices on the left, baking spices on the right, taco and chili seasonings near the top, herbs near the bottom. So these spices might actually stay where they are for a while; at least, until we get a few more and have to make a new arrangement.
How about you? Have you ever found that your household projects are related to a subtle theorem of differential geometry? Or a game you play on your phone? I’m curious to know!