The puzzle game genre of “making shapes from formulas” is currently ruled by a single designer. Five years after his last sensational foray into the genre, Iranian-born game creator Mahdi Bahrami is back with an even more impressive – and at times hair-pulling difficult – enigmatic masterpiece.
dentalAvailable this week on Steam and as an instant, DRM-free purchase for $15 (on sale for a limited time for $13.49), is arguably the coolest performance of high-level math I’ve ever seen in a PC game. In fact, his challenge is disguised in the form of a tinkerer. Give this to any young budding mathematician and watch them become addicted to what is ultimately a brilliantly disguised edutainment gem.
Axes and allies
The beauty of dental comes from how it turns the formula manipulation of X, Y and Z axes into a game mechanic. Typical teaching of mathematical formulas revolves around plotting solution results on a 2D grid to see what shapes they generate. That’s fine enough, although it requires the mathematical gibberish of a formula itself. But what if you could do this kind of thing much faster, and in 3D, by dragging shapes onto an easy-to-understand array of grids and then watching them transform into fantastic new shapes in response?
In any dental puzzle, players are given a single geometric shape (sometimes with full 3D properties, sometimes as a flat 2D polygon), and then the solution of the puzzle is shown, which is another geometric shape. Let’s start with a simple example. In the first puzzle of the game, your starting shape (which you can pick up) is one-fourth the volume of the “solution block” (which you can’t touch).
Pick up the shape with your mouse and place it on a grid made of black and white squares. To its right, a new shape will appear on a grid of taller black and white squares, and the resulting shape is twice as big on all axes – which happens to be how much bigger the black and white squares on the right are. (Mathematically, that’s a simple multiplication of the values of all axes.) You can pick up either shape, and at this point the larger one fits the shape of the solution perfectly. Pick it up and move it to a podium next to the solution and dental will scan your submission to confirm that the size, shape and curves are close enough.
The following puzzle hints how hairy dental will eventually be. The left grid is made of black and white squares, while the other grid is full of squiggly lines. Place a rectangular shape on the left grid and it will come out of the right grid with its X and Y axes in an equally serpentine shape. (Mathematically, we’re looking at a parabolic equation applied to a single axis here.)
From now on, dental only gets more intense, and the challenge potential is perhaps best summarized by the collection of shape manipulation GIFs below, created by Bahrami.
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The GIFs that follow this image contain possible spoilers for puzzle solutions, but they also clearly illustrate how different grid patterns affect all the shapes on the table. and how to lift pieces from different grids to gradually reach a solution shape.
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How to turn a 2D square into a 3D dental donut.
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Not only can you lift shapes, you can also rotate their orientation, which is crucial to achieve the correct “solution” shapes.
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One of the simpler 3D manipulation grid progressions: take a 2D shape and wrap it in a 3D donut with specific curves.
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Notice how the far right grid translates the shapes of other grids by changing elements on all three axes. By repeating this translation twice, a very unique shape is created.
If a puzzle has multiple grids on its table, you can place a shape on each of them to generate manipulated shapes on the other grids. Use your mouse to lift one of those new shapes, then move it to another grid for a recursive formula application. If one grid transforms a 2D shape into a 3D donut, you can grab the resulting donut, slide it to the donut-izing grid and distort it a lot more as a result. In addition, the exact placement of 3D shapes on the more intense grids changes which parts of their shapes are manipulated.
