Augmented triangular prism: Difference between revisions

Revision as of 14:33, 13 March 2024

Revision as of 07:41, 15 May 2024

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<math display="block"> \frac{2\sqrt{2} + 3\sqrt{3}}{12}a^3 \approx 0.669a^3. </math>

It has [[Point groups in three dimensions|three-dimensional symmetry group]] of the cyclic group <math> C_{2\mathrm{v}} </math> of order 4. Its [[dihedral angle]] can be calculated by adding the angle of an equilateral square pyramid and a regular triangular prism. The dihedral angle of an equilateral square pyramid between two adjacent triangular faces is <math display="inline"> \arccos \left(-1/3 \right) \approx 109.5^\circ </math>, and that between a triangular face and its base is <math display="inline"> \arctan \left(\sqrt{2}\right) \approx 54.7^\circ </math>. The dihedral angle of a triangular prism between two adjacent square faces isin the [[internal angle]] of an equilateral triangle <math display="inline"> \pi/3 = 60^\circ </math>, and that between square-to-triangle is <math display="inline"> \pi/2 = 90^\circ </math>. Therefore, the dihedral angle of the augmented triangular prism between square-to-triangle and triangle-to-triangle on the edge where both square pyramid and triangular prism are attached is, respectivelyfollowing:{{r|johnson}}

* The dihedral angle of a biaugmented triangular prism between two adjacent triangles is that of an equilateral square pyramid between two adjacent triangular faces, <math display="inline"> \arccos \left(-1/3 \right) \approx 109.5^\circ </math>, and is that of the interior angle of a triangular prism <math> \pi/3 = 60^\circ </math>.

<math display="block"> \begin{align}

* The dihedral angle of a biaugmented triangular prism between square and triangle is the dihedral angle of a triangular prism between the base and its lateral face, <math display="inline"> \pi/2 = 90^\circ </math>

\frac{\pi}{3} + \arccos \left(-\frac{1}{3}\right) &\approx 104.5^\circ, \\

* The dihedral angle of an equilateral square pyramid between a triangular face and its base is <math display="inline"> \arctan \left(\sqrt{2}\right) \approx 54.7^\circ </math>. The dihedral angle of a triangular prism between two adjacent square faces is the [[internal angle]] of an equilateral triangle <math display="inline"> \pi/3 = 60^\circ </math>. Therefore, the dihedral angle of a biaugmented triangular prism between a square (the lateral face of the triangular prism) and triangle (the lateral face of the equilateral square pyramid) on the edge where the equilateral square pyramid is attached to the square face of the triangular prism, and between two adjacent triangles (the lateral face of both equilateral square pyramids) on the edge where two equilateral square pyramids are attached adjacently to the triangular prism, are <math display="block"> \begin{align}

\frac{\pi}{2} + \arccos \left(-\frac{1}{3}\right) &\approx 144.5^\circ.

\frac{arctan \pi}left(\sqrt{32}\right) + \arccos \left(-\frac{1\pi}{3}\right) &\approx 104114.57^\circ, \\

\end{align} </math>

2 \arctan \left(\sqrt{2}\right) + \frac{\pi}{3} &\approx 169.4^\circ.

\end{align} </math>

</math>

== Application ==