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| - | {{Shape|Hexadecachoron|''No image''|4|16, 32, 24, 8|0|{[[Triangle|3,]][[Tetrahedron|3,]]4}|N/A|[[Line (object)|E]][[Square|E]][[Octahedron|D]]D|N/A|[[Octahedron]]|Hex|[[Tesseract]]|N/A|<xyzw>|27}}
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| - | The 16-cell is the ''dual'' of the [[tesseract]]. It is equivalent to the tesseract with vertices and cells exchanged. This is analogous with how the [[octahedron]] is the dual of the [[cube]].
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| - | == Geometry ==
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| - | === Equations ===
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| - | *Variables:
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| - | <blockquote>''l'' ⇒ length of the edges of the hexadecachoron</blockquote>
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| - | *All points (''x'', ''y'', ''z'', ''w'') that lie on the [[surcell]] of a hexadecachoron will satisfy the following equation:
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| - | <blockquote>''Unknown''</blockquote>
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| - | *The [[hypervolume]]s of a hexadecachoron are given by:
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| - | <blockquote>total edge length = 24''l''<br>
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| - | total surface area = 8sqrt(3)''l''<sup>2</sup><br>
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| - | surcell volume = 4sqrt(2)''l''<sup>3</sup>3<sup>-1</sup><br>
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| - | bulk = ''Unknown''</blockquote>
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| - | *The [[realmic]] [[cross-section]]s (''n'') of a hexadecachoron are:
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| - | <blockquote>''Unknown''</blockquote>
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| - | == Projection ==
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| - | === Vertex-first Projection ===
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| - | The vertex-first projection of the 16-cell is perhaps easiest to understand. It has an octahedral envelope, with the closest vertex lying at the center of the octahedron, with 8 tetrahedral cells surrounding it. The other 8 cells lie “behind” these 8 cells in the W direction, and meet at the opposite vertex.
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| - | <blockquote>http://tetraspace.alkaline.org/images/16-cell-01.png</blockquote>
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| - |
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| - | === Cell-first Projection ===
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| - | The cell-first projection of the 16-cell has a ''cubical'' envelope. This projection is interesting in that all the edges of the 16-cell project onto the edges of the cube and the diagonals on the faces of the cube, forming a wireframe of the cube with each face crossed. It corresponds with the two possible ways to inscribe a tetrahedron inside a cube such that the edges of the tetrahedron lie on the faces of the cube.
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| - |
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| - | <blockquote>http://tetraspace.alkaline.org/images/16-cell-02.png</blockquote>
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| - | The blue edges outline the cell closest to the viewer. Between this cell and the cubical envelope are 4 tetrahedral volumes that correspond with the 4 cells surrounding this blue cell. There are 6 cells that project onto the 6 faces of the cubical envelope (not shown here because they are being viewed edge-on). These 6 cells connect with the other side of the 16-cell, which contains the remaining 5 cells in a dual arrangement to the cells seen here. The following figure shows this opposite cell:
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| - |
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| - | <blockquote>http://tetraspace.alkaline.org/images/16-cell-03.png</blockquote>
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| - | 4D is unique in that the cell-first projection of its hypercube, which is a 3D cube, has the same envelope as the cell-first projection of its cross polytope. Only 2D has the same phenomenon, although in 2D the hypercube and the cross polytope are in fact the same object (the square).
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| - | == Construction ==
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| - | The 16-cell is so called because it consists of 16 tetrahedra joined 4 to an edge. This method of thinking about the 16-cell, however, can be difficult to visualize. It is relatively easy to understand how 4 tetrahedra sharing a common edge can be “bent” into 4D so that they each share a face with two other tetrahedra. But it is not quite so simple to imagine how 4 sets of tetrahedra attached thus can be assembled into a 16-cell.
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| - | Another method of constructing the 16-cell, which may be a bit easier to understand, is to begin with a 3D octahedron and ''tapering'' it in both the positive and negative directions along the W-axis. Consider the first case. The process of tapering is simply to stack progressively smaller octahedra on top of each other along the W-axis, until they have vanished to a single point. The tapered object is the trace formed by this process. As we taper the octahedron along the positive W-axis, each of its 8 triangular faces also grow smaller, stacking on top of the larger triangles behind it. At the end of the taper, the triangles vanish into a common apex. Thus, the tapering process forms 8 tetrahedral cells bent into the positive W-axis. This forms half of the 16-cell, with the apex being the +W vertex of the 16-cell. Now we repeat the same process in the negative W-axis, and we get another 8 tetrahedral cells in exactly the same formation, except that they are now bent in the negative W-axis, and form another apex on the opposite point, which is the -W vertex of the 16-cell. Joining these two halves together at their octahedral base, we obtain the entire 16-cell. The remaining 6 vertices of the 16-cell are precisely the 6 vertices of the octahedron we started out with.
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| - | With this understanding of the 16-cell, it is easy to see why it is the 4D equivalent of the octahedron. Just as the 2D square diamond is a 1D line segment tapered in the positive and negative Y-axis, and the octahedron is the 2D square diamond tapered in the positive and negative Z-axis, the 16-cell is the octahedron tapered in the positive and negative W-axis.
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| - | {{Polychora}}
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| - | {{Bracketope Nav|26|27|28|<[xy]zw><br>Wide hexadecachoron|<xyzw><br>Hexadecachoron|<(xy)zw><br>Dibicone|chora}}
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