{"id":836,"date":"2014-04-15T18:41:59","date_gmt":"2014-04-15T16:41:59","guid":{"rendered":"http:\/\/marctenbosch.com\/news\/?p=836"},"modified":"2023-03-02T03:46:25","modified_gmt":"2023-03-02T01:46:25","slug":"what-is-that-shape-at-the-end-of-the-trailer-hd-screenshot","status":"publish","type":"post","link":"https:\/\/marctenbosch.com\/news\/2014\/04\/what-is-that-shape-at-the-end-of-the-trailer-hd-screenshot\/","title":{"rendered":"What is that shape at the end of the trailer?"},"content":{"rendered":"\n

\"Miegakure<\/a><\/p>\n

The shape at the end of the trailer<\/a> is called a 120-cell<\/strong><\/a> (or Polydodecahedron, or Hecatonicosachoron, which sounds cooler, but a bit too hard to pronounce). It is actually modified a bit, but first let me explain some basics.<\/p>\n

You see, a 3D object has a 2D surface, whereas a 4D object has a 3D surface (an nD object has an (n-1)D surface). So while a dodecahedron has 12 faces which are pentagons, a 120-cell has 120 “faces” which are dodecahedra (called cells, since they are 3D).<\/p>\n

The 120-cell is a Convex Regular Polychoron<\/a>, the 4D analogs of the 3D Platonic Solids<\/a> (Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron). All the faces of a 3D platonic solid are the same 2D regular polygon, while all the cells of a “4D platonic solid” are the same 3D platonic solid! It’s basically building shapes out of the the most symmetrical elements each time.<\/p>\n

Interestingly, there are infinitely many regular 2D polygons (just divide the circle equally into n sides: triangle, square, pentagon, etc…), 5 regular 3D convex polyhedra, 6 regular 4D convex polychora, but in 5D or more there are only 3 types! It appears that building shapes this way gets more and more complex until it is no longer possible, save for a few very generalizable cases (the hypercube, for example).<\/p>\n

\"Leonardo<\/a><\/p>\n

But what you see in the game is not quite a 120-cell, it is actually a 120-cell with a hole inside each cell. My friend Vi Hart<\/a> [Youtube Channel<\/a>] came up with the idea to do this, inspired by drawings by Leonardo Da-Vinci<\/a>.<\/p>\n

In the drawings, a hole has been cut inside each 2D face, or rather only the edges <\/em> are visible. In the game a hole has been cut inside each 3D cell (each cell is hollow), or rather only the faces<\/em> are visible.<\/p>\n

The way this is implemented in the game engine is using our 4D Mesh Structure (a 3D mesh is made out of triangles, a 4D mesh is made out of tetrahedra).<\/p>\n

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\"Banana<\/a><\/p>\n

What you see is a 3D slice of a 4D object. While the 4D object is static, the 3D object you see transforms as the slice changes, similar to what you see in a moving slice produced by an MRI machine, but in one higher dimension (The image on the right is an MRI slicing though a banana flower).<\/p>\n

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