Elongated triangular pyramid: Difference between revisions

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{{Short description|Polyhedron7th constructedJohnson withsolid tetrahedra(7 and a triangular prismfaces)}}

{{Infobox polyhedron

| image = elongated_triangular_pyramid.png

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| rotation_group = {{math|C{{sub|3}}, [3]{{sup|+}}, (33)}}

| vertex_config = {{math|1(3{{sup|3}})<br>3(3.4{{sup|2}})<br>3(3{{sup|2}}.4{{sup|2}})}}

| dual = [[self-dual]]{{r|draghicescu}}

| properties = [[convex set|convex]]

| net = Elongated Triangular Pyramid Net.svg

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* the dihedral angle of the triangular prism between the square to its bases is <math display="inline"> \frac{\pi}{2} = 90^\circ </math>, and the dihedral angle between square-to-triangle, on the edge where tetrahedron and triangular prism are attached, is <math display="inline"> \arccos \left(\frac{1}{3}\right) + \frac{\pi}{2} \approx 160.5^\circ </math>;

* the dihedral angle of the triangular prism between two adjacent square faces is the internal angle of an equilateral triangle <math display="inline"> \frac{\pi}{3} = 60^\circ </math>.

=== Dual polyhedron ===

Topologically, the elongated triangular pyramid is its own dual. Geometrically, the dual has seven irregular faces: one equilateral triangle, three isosceles triangles and three isosceles trapezoids.

{| class=wikitable width=320

|- valign=top

!Dual elongated triangular pyramid

!Net of dual

|- valign=top

|[[File:Dual elongated triangular pyramid.png|160px]]

|[[File:Dual elongated triangular pyramid net.png|160px]]

|}

==Related polyhedra and honeycombs==

The elongated triangular pyramid can form a [[tessellation of space]] with [[square pyramid]]s and/or [[Octahedron|octahedra]].<ref>{{Cite web|url=http://woodenpolyhedra.web.fc2.com/J7.html|title=J7 honeycomb}}</ref>

== References ==

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| doi = 10.1016/0016-0032(71)90071-8

| mr = 290245

}}</ref>

<ref name="draghicescu">{{cite book

| last = Draghicescu | first = Mircea

| contribution = Dual Models: One Shape to Make Them All

| contribution-url = https://archive.bridgesmathart.org/2016/bridges2016

| editor-first1 = Eva | editor-last1 = Torrence

| editor-first2 = Bruce | editor-last2 = Torrence

| editor-first3 = Carlo H. | editor-last3 = Séquin

| editor-first4 = Douglas | editor-last4 = McKenna

| editor-first5 = Kristóf | editor-last5 = Fenyvesi

| editor-first6 = Reza | editor-last6 = Sarhangi

| title = Bridges Finland: Mathematics, Music, Art, Architecture, Education, Culture

| url = https://archive.bridgesmathart.org/2016/frontmatter.pdf

| pages = 635–640

}}</ref>