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<blockquote data-quote="Conaill" data-source="post: 2093197" data-attributes="member: 1264"><p>As Fieari already alluded to, it's not quite that easy.</p><p></p><p>Actually, if you want a 4D "die", you don't want just any extrusion into 4D. What you really want is a 4D equivalent of the <em>platonic solids</em>, aka regular polyhedra. So what you're after are the <em>regular polychora</em>. Cool name, huh?</p><p></p><p>A polychoron is a 4D polytope, and a polytope is just a generalisation of the sequence point - line segment - polygon - polyhedron - polychoron. The "surface" of each regular polychora consist of regular polyhedra. Luckily, you can find more info on these beasties on <a href="http://mathworld.wolfram.com/RegularPolychoron.html" target="_blank">Mathworld</a>. Apparently, there are only 16 regular polychora (just like there's only 5 platonic solids: d4, d6, d8, d12, d20). </p><p></p><p>For example, the hypercube is constructed by joining all the vertices of a cube with the corresponding vertices of a second cube offset in a 4th dimension. This polychoron is bounded by 8 cubes: the two on each end, and 6 more formed by connecting each side of the first cube to the second cube.</p><p></p><p>The other easy regular polychoron is the pentatope: take a tetrahedron (d4), and add one extra vertex equidistant from the other four nodes in 4D. It's pretty easy to see that the pentatope consists of 5 vertices, 10 edges, 10 triangles, and 5 tetrahedra...</p><p></p><p>Now... of course it all depends on how you want to roll these 4D dice! I would recommend a 4D table top, which means that the dice will come to rest on one of their polyhedra, rather than one of their polygons. Besides, a 4D table top also provides more space for sodas, charactersheets, etc. <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":)" title="Smile :)" data-smilie="1"data-shortname=":)" /></p></blockquote><p></p>
[QUOTE="Conaill, post: 2093197, member: 1264"] As Fieari already alluded to, it's not quite that easy. Actually, if you want a 4D "die", you don't want just any extrusion into 4D. What you really want is a 4D equivalent of the [i]platonic solids[/i], aka regular polyhedra. So what you're after are the [i]regular polychora[/i]. Cool name, huh? A polychoron is a 4D polytope, and a polytope is just a generalisation of the sequence point - line segment - polygon - polyhedron - polychoron. The "surface" of each regular polychora consist of regular polyhedra. Luckily, you can find more info on these beasties on [url=http://mathworld.wolfram.com/RegularPolychoron.html]Mathworld[/url]. Apparently, there are only 16 regular polychora (just like there's only 5 platonic solids: d4, d6, d8, d12, d20). For example, the hypercube is constructed by joining all the vertices of a cube with the corresponding vertices of a second cube offset in a 4th dimension. This polychoron is bounded by 8 cubes: the two on each end, and 6 more formed by connecting each side of the first cube to the second cube. The other easy regular polychoron is the pentatope: take a tetrahedron (d4), and add one extra vertex equidistant from the other four nodes in 4D. It's pretty easy to see that the pentatope consists of 5 vertices, 10 edges, 10 triangles, and 5 tetrahedra... Now... of course it all depends on how you want to roll these 4D dice! I would recommend a 4D table top, which means that the dice will come to rest on one of their polyhedra, rather than one of their polygons. Besides, a 4D table top also provides more space for sodas, charactersheets, etc. :) [/QUOTE]
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