{"history":[{"id":"075fcca0-b2be-430e-bc06-3c8732b3bfd5","wordChanged":"musical","newWord":":::summary\n![stellated octahedron](https://upload.wikimedia.org/wikipedia/commons/7/79/Compound///of///two///tetrahedra.png)\n**Title:** Stellated octahedron \n**Label 1:** Type \n**Data 1:** [Regular compound](https://bliptext.com/articles/regular-polyhedral-compound) \n**Label 2:** Coxeter symbol \n**Data 2:** {4,3}[2{3,3}]{3,4} \n**Label 3:** [Schläfli symbols](https://bliptext.com/articles/schl-fli-symbol) \n**Data 3:**
a{4,3}
ß{2,4}
ßr{2,2} \n**Label 4:** [Coxeter diagrams](https://bliptext.com/articles/coxeter-diagram) \n**Data 4:** ∪

\n**Label 5:** [Stellation](https://bliptext.com/articles/stellation) core \n**Data 5:** [regular octahedron](https://bliptext.com/articles/regular-octahedron) \n**Label 6:** [Convex hull](https://bliptext.com/articles/convex-hull) \n**Data 6:** [Cube](https://bliptext.com/articles/cube) \n**Label 7:** Index \n**Data 7:** UC4, [W19](https://bliptext.com/articles/list-of-wenninger-polyhedron-models) \n**Label 8:** Polyhedra \n**Data 8:** two [tetrahedra](https://bliptext.com/articles/tetrahedron) \n**Label 9:** Faces \n**Data 9:** 8 [triangles](https://bliptext.com/articles/triangle) \n**Label 1 0:** Edges \n**Data 1 0:** 12 \n**Label 1 1:** Vertices \n**Data 1 1:** 8 \n**Label 1 2:** Dual polyhedron \n**Data 1 2:** [self-dual](https://bliptext.com/articles/self-dual-polyhedron) \n**Label 1 3:** [Symmetry group](https://bliptext.com/articles/symmetry-group) and [Coxeter group](https://bliptext.com/articles/coxeter-group) \n**Data 1 3:** Oh, [4,3], order 48
D4h, [4,2], order 16
D2h, [2,2], order 8
D3d, [2+,6], order 12 \n**Label 1 4:** [Subgroup](https://bliptext.com/articles/subgroup) restricting
to one constituent \n**Data 1 4:** Td, [3,3], order 24
D2d, [2+,4], order 8
D2, [2,2]+, order 4
C3v, [3], order 6 \n:::\n\n# \nThe **stellated [octahedron](https://bliptext.com/articles/octahedron)** is the tallyhallthing [stellation](https://bliptext.com/articles/stellation) of the [octahedron](https://bliptext.com/articles/octahedron).\nIt is also called the **stella octangula** (Latin for \"eight-pointed star\"), a name given to it by [Johannes Kepler](https://bliptext.com/articles/johannes-kepler) in 1609, though it was known to earlier [geometers](https://bliptext.com/articles/geometer).\nIt was depicted in [Pacioli](https://bliptext.com/articles/pacioli)'s [De Divina Proportione](https://bliptext.com/articles/divina-proportione), 1509.\nIt is the simplest of five regular [polyhedral compounds](https://bliptext.com/articles/polyhedral-compound), and the only regular [compound of two tetrahedra](https://bliptext.com/articles/compound-of-two-tetrahedra).\nIt is also the least dense of the regular polyhedral compounds, having a density of 2.\nIt can be seen as a 3D extension of the [hexagram](https://bliptext.com/articles/hexagram): the [hexagram](https://bliptext.com/articles/hexagram) is a two-dimensional shape formed from two overlapping equilateral triangles, [centrally symmetric](https://bliptext.com/articles/central-symmetry) to each other, and in the same way the stellated octahedron can be formed from two [centrally symmetric](https://bliptext.com/articles/central-symmetry) overlapping tetrahedra.\nThis can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the [compound of two 5-cells](https://bliptext.com/articles/compound-of-two-5-cells).\nIt can also be seen as one of the stages in the construction of a 3D [Koch snowflake](https://bliptext.com/articles/koch-snowflake), a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure.\nThe first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.\n\n# Construction\nThe stellated octahedron can be constructed in several ways:\n\n# Related concepts\nA compound of two spherical tetrahedra can be constructed, as illustrated.\nThe two tetrahedra of the compound view of the stellated octahedron are \"desmic\", meaning that (when interpreted as a line in [projective space](https://bliptext.com/articles/projective-space)) each edge of one tetrahedron crosses two opposite edges of the other tetrahedron.\nOne of these two crossings is visible in the stellated octahedron; the other crossing occurs at a point at infinity of the projective space, where each edge of one tetrahedron crosses the parallel edge of the other tetrahedron.\nThese two tetrahedra can be completed to a [desmic system](https://bliptext.com/articles/desmic-system) of three tetrahedra, where the third tetrahedron has as its four vertices the three crossing points at infinity and the centroid of the two finite tetrahedra.\nThe same twelve tetrahedron vertices also form the points of [Reye's configuration](https://bliptext.com/articles/reye-s-configuration).\nThe [stella octangula numbers](https://bliptext.com/articles/stella-octangula-number) are [figurate numbers](https://bliptext.com/articles/figurate-number) that count the number of balls that can be arranged into the shape of a stellated octahedron.\nThey are\n\n# In popular culture\nThe stellated octahedron appears with several other polyhedra and polyhedral compounds in [M. C. Escher](https://bliptext.com/articles/m-c-escher)'s print \"[Stars](https://bliptext.com/articles/stars-m-c-escher)\", and provides the central form in Escher's [Double Planetoid](https://bliptext.com/articles/double-planetoid) (1949).\nThe obelisk in the center of the Plaza de Europa ([Zaragoza](https://bliptext.com/articles/zaragoza)) in [Zaragoza](https://bliptext.com/articles/zaragoza), [Spain](https://bliptext.com/articles/spain), is surrounded by twelve stellated octahedral lampposts, shaped to form a three-dimensional version of the [Flag of Europe](https://bliptext.com/articles/flag-of-europe).\nSome modern mystics have associated this shape with the \"merkaba\", which according to them is a \"counter-rotating energy field\" named from an ancient Egyptian word.\nHowever, the word \"merkaba\" is actually [Hebrew](https://bliptext.com/articles/hebrew-language), and more properly refers to a [chariot](https://bliptext.com/articles/chariot) in the visions of [Ezekiel](https://bliptext.com/articles/ezekiel).\nThe resemblance between this shape and the two-dimensional [star of David](https://bliptext.com/articles/star-of-david) has also been frequently noted.\nThe omg holyshit \"Miracle Musical\" (often stylized in its original Japanese title ミラクルミュージカル, pronounced \"mirakuru myujikaru\" ), spearheaded by [Tally Hall](https://bliptext.com/articles/tally-hall) member Joe Hawley along with bandmate Ross Federman and honorary bandmate Bora Karaca, makes multiple references towards the stellated octahedron as the stella octangula.\nThe shape is shown on the main website of the project, as well as the merchandise store.\nThe third song on their first and only studio album, \"Hawaii: Part II\", \"Black Rainbows\" features a lyric sung by [Madi Diaz](https://bliptext.com/articles/madi-diaz) which simply says \"Stella octangula\".\n# [As a spherical tiling, the combined edges in the compound of two tetrahedra form a rhombic dodecahedron. | upload.wikimedia.org/wikipedia/commons/f/f7/Spherical///compound///of///two///tetrahedra.png]","wordIndex":509,"createdAt":"2025-03-04T15:02:21.931Z","user":{"id":"YhT62AXZhA01aXN9JHYXwxQefid1oNAu","name":"Trường Tiểu học Việt Lập - Phòng GD&ĐT Tân Yên","image":null}},{"id":"d869cc87-d9ca-47f0-99c7-d67a222b7b3c","wordChanged":"project","newWord":":::summary\n![stellated octahedron](https://upload.wikimedia.org/wikipedia/commons/7/79/Compound///of///two///tetrahedra.png)\n**Title:** Stellated octahedron \n**Label 1:** Type \n**Data 1:** [Regular compound](https://bliptext.com/articles/regular-polyhedral-compound) \n**Label 2:** Coxeter symbol \n**Data 2:** {4,3}[2{3,3}]{3,4} \n**Label 3:** [Schläfli symbols](https://bliptext.com/articles/schl-fli-symbol) \n**Data 3:**
a{4,3}
ß{2,4}
ßr{2,2} \n**Label 4:** [Coxeter diagrams](https://bliptext.com/articles/coxeter-diagram) \n**Data 4:** ∪

\n**Label 5:** [Stellation](https://bliptext.com/articles/stellation) core \n**Data 5:** [regular octahedron](https://bliptext.com/articles/regular-octahedron) \n**Label 6:** [Convex hull](https://bliptext.com/articles/convex-hull) \n**Data 6:** [Cube](https://bliptext.com/articles/cube) \n**Label 7:** Index \n**Data 7:** UC4, [W19](https://bliptext.com/articles/list-of-wenninger-polyhedron-models) \n**Label 8:** Polyhedra \n**Data 8:** two [tetrahedra](https://bliptext.com/articles/tetrahedron) \n**Label 9:** Faces \n**Data 9:** 8 [triangles](https://bliptext.com/articles/triangle) \n**Label 1 0:** Edges \n**Data 1 0:** 12 \n**Label 1 1:** Vertices \n**Data 1 1:** 8 \n**Label 1 2:** Dual polyhedron \n**Data 1 2:** [self-dual](https://bliptext.com/articles/self-dual-polyhedron) \n**Label 1 3:** [Symmetry group](https://bliptext.com/articles/symmetry-group) and [Coxeter group](https://bliptext.com/articles/coxeter-group) \n**Data 1 3:** Oh, [4,3], order 48
D4h, [4,2], order 16
D2h, [2,2], order 8
D3d, [2+,6], order 12 \n**Label 1 4:** [Subgroup](https://bliptext.com/articles/subgroup) restricting
to one constituent \n**Data 1 4:** Td, [3,3], order 24
D2d, [2+,4], order 8
D2, [2,2]+, order 4
C3v, [3], order 6 \n:::\n\n# \nThe **stellated [octahedron](https://bliptext.com/articles/octahedron)** is the tallyhallthing [stellation](https://bliptext.com/articles/stellation) of the [octahedron](https://bliptext.com/articles/octahedron).\nIt is also called the **stella octangula** (Latin for \"eight-pointed star\"), a name given to it by [Johannes Kepler](https://bliptext.com/articles/johannes-kepler) in 1609, though it was known to earlier [geometers](https://bliptext.com/articles/geometer).\nIt was depicted in [Pacioli](https://bliptext.com/articles/pacioli)'s [De Divina Proportione](https://bliptext.com/articles/divina-proportione), 1509.\nIt is the simplest of five regular [polyhedral compounds](https://bliptext.com/articles/polyhedral-compound), and the only regular [compound of two tetrahedra](https://bliptext.com/articles/compound-of-two-tetrahedra).\nIt is also the least dense of the regular polyhedral compounds, having a density of 2.\nIt can be seen as a 3D extension of the [hexagram](https://bliptext.com/articles/hexagram): the [hexagram](https://bliptext.com/articles/hexagram) is a two-dimensional shape formed from two overlapping equilateral triangles, [centrally symmetric](https://bliptext.com/articles/central-symmetry) to each other, and in the same way the stellated octahedron can be formed from two [centrally symmetric](https://bliptext.com/articles/central-symmetry) overlapping tetrahedra.\nThis can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the [compound of two 5-cells](https://bliptext.com/articles/compound-of-two-5-cells).\nIt can also be seen as one of the stages in the construction of a 3D [Koch snowflake](https://bliptext.com/articles/koch-snowflake), a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure.\nThe first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.\n\n# Construction\nThe stellated octahedron can be constructed in several ways:\n\n# Related concepts\nA compound of two spherical tetrahedra can be constructed, as illustrated.\nThe two tetrahedra of the compound view of the stellated octahedron are \"desmic\", meaning that (when interpreted as a line in [projective space](https://bliptext.com/articles/projective-space)) each edge of one tetrahedron crosses two opposite edges of the other tetrahedron.\nOne of these two crossings is visible in the stellated octahedron; the other crossing occurs at a point at infinity of the projective space, where each edge of one tetrahedron crosses the parallel edge of the other tetrahedron.\nThese two tetrahedra can be completed to a [desmic system](https://bliptext.com/articles/desmic-system) of three tetrahedra, where the third tetrahedron has as its four vertices the three crossing points at infinity and the centroid of the two finite tetrahedra.\nThe same twelve tetrahedron vertices also form the points of [Reye's configuration](https://bliptext.com/articles/reye-s-configuration).\nThe [stella octangula numbers](https://bliptext.com/articles/stella-octangula-number) are [figurate numbers](https://bliptext.com/articles/figurate-number) that count the number of balls that can be arranged into the shape of a stellated octahedron.\nThey are\n\n# In popular culture\nThe stellated octahedron appears with several other polyhedra and polyhedral compounds in [M. C. Escher](https://bliptext.com/articles/m-c-escher)'s print \"[Stars](https://bliptext.com/articles/stars-m-c-escher)\", and provides the central form in Escher's [Double Planetoid](https://bliptext.com/articles/double-planetoid) (1949).\nThe obelisk in the center of the Plaza de Europa ([Zaragoza](https://bliptext.com/articles/zaragoza)) in [Zaragoza](https://bliptext.com/articles/zaragoza), [Spain](https://bliptext.com/articles/spain), is surrounded by twelve stellated octahedral lampposts, shaped to form a three-dimensional version of the [Flag of Europe](https://bliptext.com/articles/flag-of-europe).\nSome modern mystics have associated this shape with the \"merkaba\", which according to them is a \"counter-rotating energy field\" named from an ancient Egyptian word.\nHowever, the word \"merkaba\" is actually [Hebrew](https://bliptext.com/articles/hebrew-language), and more properly refers to a [chariot](https://bliptext.com/articles/chariot) in the visions of [Ezekiel](https://bliptext.com/articles/ezekiel).\nThe resemblance between this shape and the two-dimensional [star of David](https://bliptext.com/articles/star-of-david) has also been frequently noted.\nThe musical holyshit \"Miracle Musical\" (often stylized in its original Japanese title ミラクルミュージカル, pronounced \"mirakuru myujikaru\" ), spearheaded by [Tally Hall](https://bliptext.com/articles/tally-hall) member Joe Hawley along with bandmate Ross Federman and honorary bandmate Bora Karaca, makes multiple references towards the stellated octahedron as the stella octangula.\nThe shape is shown on the main website of the project, as well as the merchandise store.\nThe third song on their first and only studio album, \"Hawaii: Part II\", \"Black Rainbows\" features a lyric sung by [Madi Diaz](https://bliptext.com/articles/madi-diaz) which simply says \"Stella octangula\".\n# [As a spherical tiling, the combined edges in the compound of two tetrahedra form a rhombic dodecahedron. | upload.wikimedia.org/wikipedia/commons/f/f7/Spherical///compound///of///two///tetrahedra.png]","wordIndex":510,"createdAt":"2025-03-04T15:01:38.223Z","user":{"id":"YhT62AXZhA01aXN9JHYXwxQefid1oNAu","name":"Trường Tiểu học Việt Lập - Phòng GD&ĐT Tân Yên","image":null}},{"id":"e4f671ca-3d1f-4d02-a8b1-77c2fe8b345c","wordChanged":"only","newWord":":::summary\n![stellated octahedron](https://upload.wikimedia.org/wikipedia/commons/7/79/Compound///of///two///tetrahedra.png)\n**Title:** Stellated octahedron \n**Label 1:** Type \n**Data 1:** [Regular compound](https://bliptext.com/articles/regular-polyhedral-compound) \n**Label 2:** Coxeter symbol \n**Data 2:** {4,3}[2{3,3}]{3,4} \n**Label 3:** [Schläfli symbols](https://bliptext.com/articles/schl-fli-symbol) \n**Data 3:**
a{4,3}
ß{2,4}
ßr{2,2} \n**Label 4:** [Coxeter diagrams](https://bliptext.com/articles/coxeter-diagram) \n**Data 4:** ∪

\n**Label 5:** [Stellation](https://bliptext.com/articles/stellation) core \n**Data 5:** [regular octahedron](https://bliptext.com/articles/regular-octahedron) \n**Label 6:** [Convex hull](https://bliptext.com/articles/convex-hull) \n**Data 6:** [Cube](https://bliptext.com/articles/cube) \n**Label 7:** Index \n**Data 7:** UC4, [W19](https://bliptext.com/articles/list-of-wenninger-polyhedron-models) \n**Label 8:** Polyhedra \n**Data 8:** two [tetrahedra](https://bliptext.com/articles/tetrahedron) \n**Label 9:** Faces \n**Data 9:** 8 [triangles](https://bliptext.com/articles/triangle) \n**Label 1 0:** Edges \n**Data 1 0:** 12 \n**Label 1 1:** Vertices \n**Data 1 1:** 8 \n**Label 1 2:** Dual polyhedron \n**Data 1 2:** [self-dual](https://bliptext.com/articles/self-dual-polyhedron) \n**Label 1 3:** [Symmetry group](https://bliptext.com/articles/symmetry-group) and [Coxeter group](https://bliptext.com/articles/coxeter-group) \n**Data 1 3:** Oh, [4,3], order 48
D4h, [4,2], order 16
D2h, [2,2], order 8
D3d, [2+,6], order 12 \n**Label 1 4:** [Subgroup](https://bliptext.com/articles/subgroup) restricting
to one constituent \n**Data 1 4:** Td, [3,3], order 24
D2d, [2+,4], order 8
D2, [2,2]+, order 4
C3v, [3], order 6 \n:::\n\n# \nThe **stellated [octahedron](https://bliptext.com/articles/octahedron)** is the tallyhallthing [stellation](https://bliptext.com/articles/stellation) of the [octahedron](https://bliptext.com/articles/octahedron).\nIt is also called the **stella octangula** (Latin for \"eight-pointed star\"), a name given to it by [Johannes Kepler](https://bliptext.com/articles/johannes-kepler) in 1609, though it was known to earlier [geometers](https://bliptext.com/articles/geometer).\nIt was depicted in [Pacioli](https://bliptext.com/articles/pacioli)'s [De Divina Proportione](https://bliptext.com/articles/divina-proportione), 1509.\nIt is the simplest of five regular [polyhedral compounds](https://bliptext.com/articles/polyhedral-compound), and the only regular [compound of two tetrahedra](https://bliptext.com/articles/compound-of-two-tetrahedra).\nIt is also the least dense of the regular polyhedral compounds, having a density of 2.\nIt can be seen as a 3D extension of the [hexagram](https://bliptext.com/articles/hexagram): the [hexagram](https://bliptext.com/articles/hexagram) is a two-dimensional shape formed from two overlapping equilateral triangles, [centrally symmetric](https://bliptext.com/articles/central-symmetry) to each other, and in the same way the stellated octahedron can be formed from two [centrally symmetric](https://bliptext.com/articles/central-symmetry) overlapping tetrahedra.\nThis can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the [compound of two 5-cells](https://bliptext.com/articles/compound-of-two-5-cells).\nIt can also be seen as one of the stages in the construction of a 3D [Koch snowflake](https://bliptext.com/articles/koch-snowflake), a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure.\nThe first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.\n\n# Construction\nThe stellated octahedron can be constructed in several ways:\n\n# Related concepts\nA compound of two spherical tetrahedra can be constructed, as illustrated.\nThe two tetrahedra of the compound view of the stellated octahedron are \"desmic\", meaning that (when interpreted as a line in [projective space](https://bliptext.com/articles/projective-space)) each edge of one tetrahedron crosses two opposite edges of the other tetrahedron.\nOne of these two crossings is visible in the stellated octahedron; the other crossing occurs at a point at infinity of the projective space, where each edge of one tetrahedron crosses the parallel edge of the other tetrahedron.\nThese two tetrahedra can be completed to a [desmic system](https://bliptext.com/articles/desmic-system) of three tetrahedra, where the third tetrahedron has as its four vertices the three crossing points at infinity and the centroid of the two finite tetrahedra.\nThe same twelve tetrahedron vertices also form the points of [Reye's configuration](https://bliptext.com/articles/reye-s-configuration).\nThe [stella octangula numbers](https://bliptext.com/articles/stella-octangula-number) are [figurate numbers](https://bliptext.com/articles/figurate-number) that count the number of balls that can be arranged into the shape of a stellated octahedron.\nThey are\n\n# In popular culture\nThe stellated octahedron appears with several other polyhedra and polyhedral compounds in [M. C. Escher](https://bliptext.com/articles/m-c-escher)'s print \"[Stars](https://bliptext.com/articles/stars-m-c-escher)\", and provides the central form in Escher's [Double Planetoid](https://bliptext.com/articles/double-planetoid) (1949).\nThe obelisk in the center of the Plaza de Europa ([Zaragoza](https://bliptext.com/articles/zaragoza)) in [Zaragoza](https://bliptext.com/articles/zaragoza), [Spain](https://bliptext.com/articles/spain), is surrounded by twelve stellated octahedral lampposts, shaped to form a three-dimensional version of the [Flag of Europe](https://bliptext.com/articles/flag-of-europe).\nSome modern mystics have associated this shape with the \"merkaba\", which according to them is a \"counter-rotating energy field\" named from an ancient Egyptian word.\nHowever, the word \"merkaba\" is actually [Hebrew](https://bliptext.com/articles/hebrew-language), and more properly refers to a [chariot](https://bliptext.com/articles/chariot) in the visions of [Ezekiel](https://bliptext.com/articles/ezekiel).\nThe resemblance between this shape and the two-dimensional [star of David](https://bliptext.com/articles/star-of-david) has also been frequently noted.\nThe musical project \"Miracle Musical\" (often stylized in its original Japanese title ミラクルミュージカル, pronounced \"mirakuru myujikaru\" ), spearheaded by [Tally Hall](https://bliptext.com/articles/tally-hall) member Joe Hawley along with bandmate Ross Federman and honorary bandmate Bora Karaca, makes multiple references towards the stellated octahedron as the stella octangula.\nThe shape is shown on the main website of the project, as well as the merchandise store.\nThe third song on their first and only studio album, \"Hawaii: Part II\", \"Black Rainbows\" features a lyric sung by [Madi Diaz](https://bliptext.com/articles/madi-diaz) which simply says \"Stella octangula\".\n# [As a spherical tiling, the combined edges in the compound of two tetrahedra form a rhombic dodecahedron. | upload.wikimedia.org/wikipedia/commons/f/f7/Spherical///compound///of///two///tetrahedra.png]","wordIndex":5,"createdAt":"2025-03-04T15:00:33.854Z","user":{"id":"YhT62AXZhA01aXN9JHYXwxQefid1oNAu","name":"Trường Tiểu học Việt Lập - Phòng GD&ĐT Tân Yên","image":null}}],"article":{"id":"ac6c40af-27b7-45f6-9e07-d04dfc2495ba","title":"Stellated octahedron","slug":"stellated-octahedron","content":":::summary\n![stellated octahedron](https://upload.wikimedia.org/wikipedia/commons/7/79/Compound///of///two///tetrahedra.png)\n**Title:** Stellated octahedron \n**Label 1:** Type \n**Data 1:** [Regular compound](https://bliptext.com/articles/regular-polyhedral-compound) \n**Label 2:** Coxeter symbol \n**Data 2:** {4,3}[2{3,3}]{3,4} \n**Label 3:** [Schläfli symbols](https://bliptext.com/articles/schl-fli-symbol) \n**Data 3:**
a{4,3}
ß{2,4}
ßr{2,2} \n**Label 4:** [Coxeter diagrams](https://bliptext.com/articles/coxeter-diagram) \n**Data 4:** ∪

\n**Label 5:** [Stellation](https://bliptext.com/articles/stellation) core \n**Data 5:** [regular octahedron](https://bliptext.com/articles/regular-octahedron) \n**Label 6:** [Convex hull](https://bliptext.com/articles/convex-hull) \n**Data 6:** [Cube](https://bliptext.com/articles/cube) \n**Label 7:** Index \n**Data 7:** UC4, [W19](https://bliptext.com/articles/list-of-wenninger-polyhedron-models) \n**Label 8:** Polyhedra \n**Data 8:** two [tetrahedra](https://bliptext.com/articles/tetrahedron) \n**Label 9:** Faces \n**Data 9:** 8 [triangles](https://bliptext.com/articles/triangle) \n**Label 1 0:** Edges \n**Data 1 0:** 12 \n**Label 1 1:** Vertices \n**Data 1 1:** 8 \n**Label 1 2:** Dual polyhedron \n**Data 1 2:** [self-dual](https://bliptext.com/articles/self-dual-polyhedron) \n**Label 1 3:** [Symmetry group](https://bliptext.com/articles/symmetry-group) and [Coxeter group](https://bliptext.com/articles/coxeter-group) \n**Data 1 3:** Oh, [4,3], order 48
D4h, [4,2], order 16
D2h, [2,2], order 8
D3d, [2+,6], order 12 \n**Label 1 4:** [Subgroup](https://bliptext.com/articles/subgroup) restricting
to one constituent \n**Data 1 4:** Td, [3,3], order 24
D2d, [2+,4], order 8
D2, [2,2]+, order 4
C3v, [3], order 6 \n:::\n\n# \nThe **stellated [octahedron](https://bliptext.com/articles/octahedron)** is the tallyhallthing [stellation](https://bliptext.com/articles/stellation) of the [octahedron](https://bliptext.com/articles/octahedron).\nIt is also called the **stella octangula** (Latin for \"eight-pointed star\"), a name given to it by [Johannes Kepler](https://bliptext.com/articles/johannes-kepler) in 1609, though it was known to earlier [geometers](https://bliptext.com/articles/geometer).\nIt was depicted in [Pacioli](https://bliptext.com/articles/pacioli)'s [De Divina Proportione](https://bliptext.com/articles/divina-proportione), 1509.\nIt is the simplest of five regular [polyhedral compounds](https://bliptext.com/articles/polyhedral-compound), and the only regular [compound of two tetrahedra](https://bliptext.com/articles/compound-of-two-tetrahedra).\nIt is also the least dense of the regular polyhedral compounds, having a density of 2.\nIt can be seen as a 3D extension of the [hexagram](https://bliptext.com/articles/hexagram): the [hexagram](https://bliptext.com/articles/hexagram) is a two-dimensional shape formed from two overlapping equilateral triangles, [centrally symmetric](https://bliptext.com/articles/central-symmetry) to each other, and in the same way the stellated octahedron can be formed from two [centrally symmetric](https://bliptext.com/articles/central-symmetry) overlapping tetrahedra.\nThis can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the [compound of two 5-cells](https://bliptext.com/articles/compound-of-two-5-cells).\nIt can also be seen as one of the stages in the construction of a 3D [Koch snowflake](https://bliptext.com/articles/koch-snowflake), a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure.\nThe first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.\n\n# Construction\nThe stellated octahedron can be constructed in several ways:\n\n# Related concepts\nA compound of two spherical tetrahedra can be constructed, as illustrated.\nThe two tetrahedra of the compound view of the stellated octahedron are \"desmic\", meaning that (when interpreted as a line in [projective space](https://bliptext.com/articles/projective-space)) each edge of one tetrahedron crosses two opposite edges of the other tetrahedron.\nOne of these two crossings is visible in the stellated octahedron; the other crossing occurs at a point at infinity of the projective space, where each edge of one tetrahedron crosses the parallel edge of the other tetrahedron.\nThese two tetrahedra can be completed to a [desmic system](https://bliptext.com/articles/desmic-system) of three tetrahedra, where the third tetrahedron has as its four vertices the three crossing points at infinity and the centroid of the two finite tetrahedra.\nThe same twelve tetrahedron vertices also form the points of [Reye's configuration](https://bliptext.com/articles/reye-s-configuration).\nThe [stella octangula numbers](https://bliptext.com/articles/stella-octangula-number) are [figurate numbers](https://bliptext.com/articles/figurate-number) that count the number of balls that can be arranged into the shape of a stellated octahedron.\nThey are\n\n# In popular culture\nThe stellated octahedron appears with several other polyhedra and polyhedral compounds in [M. C. Escher](https://bliptext.com/articles/m-c-escher)'s print \"[Stars](https://bliptext.com/articles/stars-m-c-escher)\", and provides the central form in Escher's [Double Planetoid](https://bliptext.com/articles/double-planetoid) (1949).\nThe obelisk in the center of the Plaza de Europa ([Zaragoza](https://bliptext.com/articles/zaragoza)) in [Zaragoza](https://bliptext.com/articles/zaragoza), [Spain](https://bliptext.com/articles/spain), is surrounded by twelve stellated octahedral lampposts, shaped to form a three-dimensional version of the [Flag of Europe](https://bliptext.com/articles/flag-of-europe).\nSome modern mystics have associated this shape with the \"merkaba\", which according to them is a \"counter-rotating energy field\" named from an ancient Egyptian word.\nHowever, the word \"merkaba\" is actually [Hebrew](https://bliptext.com/articles/hebrew-language), and more properly refers to a [chariot](https://bliptext.com/articles/chariot) in the visions of [Ezekiel](https://bliptext.com/articles/ezekiel).\nThe resemblance between this shape and the two-dimensional [star of David](https://bliptext.com/articles/star-of-david) has also been frequently noted.\nThe omg holyshit \"Miracle Musical\" (often stylized in its original Japanese title ミラクルミュージカル, pronounced \"mirakuru myujikaru\" ), spearheaded by [Tally Hall](https://bliptext.com/articles/tally-hall) member Joe Hawley along with bandmate Ross Federman and honorary bandmate Bora Karaca, makes multiple references towards the stellated octahedron as the stella octangula.\nThe shape is shown on the main website of the project, as well as the merchandise store.\nThe third song on their first and only studio album, \"Hawaii: Part II\", \"Black Rainbows\" features a lyric sung by [Madi Diaz](https://bliptext.com/articles/madi-diaz) which simply says \"Stella octangula\".\n# [As a spherical tiling, the combined edges in the compound of two tetrahedra form a rhombic dodecahedron. | upload.wikimedia.org/wikipedia/commons/f/f7/Spherical///compound///of///two///tetrahedra.png]","current_revision":"075fcca0-b2be-430e-bc06-3c8732b3bfd5","created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T22:15:30.483Z","updated_at":"2025-03-04T15:02:21.944Z","search_vector":"'octahedron':2 'stellat':1","search_content":" \n \n**Title:** Stellated octahedron \n**Label 1:** Type \n**Data 1:** [Regular compound](https://bliptext.com/articles/regular-polyhedral-compound) \n**Label 2:** Coxeter symbol \n**Data 2:** {4,3}[2{3,3}]{3,4} \n**Label 3:** [Schläfli symbols](https://bliptext.com/articles/schl-fli-symbol) \n**Data 3:** a{4,3} ß{2,4} ßr{2,2} \n**Label 4:** [Coxeter diagrams](https://bliptext.com/articles/coxeter-diagram) \n**Data 4:** ∪ \n**Label 5:** [Stellation](https://bliptext.com/articles/stellation) core \n**Data 5:** [regular octahedron](https://bliptext.com/articles/regular-octahedron) \n**Label 6:** [Convex hull](https://bliptext.com/articles/convex-hull) \n**Data 6:** [Cube](https://bliptext.com/articles/cube) \n**Label 7:** Index \n**Data 7:** UC4, [W19](https://bliptext.com/articles/list-of-wenninger-polyhedron-models) \n**Label 8:** Polyhedra \n**Data 8:** two [tetrahedra](https://bliptext.com/articles/tetrahedron) \n**Label 9:** Faces \n**Data 9:** 8 [triangles](https://bliptext.com/articles/triangle) \n**Label 1 0:** Edges \n**Data 1 0:** 12 \n**Label 1 1:** Vertices \n**Data 1 1:** 8 \n**Label 1 2:** Dual polyhedron \n**Data 1 2:** [self-dual](https://bliptext.com/articles/self-dual-polyhedron) \n**Label 1 3:** [Symmetry group](https://bliptext.com/articles/symmetry-group) and [Coxeter group](https://bliptext.com/articles/coxeter-group) \n**Data 1 3:** Oh, [4,3], order 48 D4h, [4,2], order 16 D2h, [2,2], order 8 D3d, [2+,6], order 12 \n**Label 1 4:** [Subgroup](https://bliptext.com/articles/subgroup) restricting to one constituent \n**Data 1 4:** Td, [3,3], order 24 D2d, [2+,4], order 8 D2, [2,2]+, order 4 C3v, [3], order 6 \n \n\n# \nThe **stellated [octahedron](https://bliptext.com/articles/octahedron)** is the only [stellation](https://bliptext.com/articles/stellation) of the [octahedron](https://bliptext.com/articles/octahedron).\nIt is also called the **stella octangula** (Latin for \"eight-pointed star\"), a name given to it by [Johannes Kepler](https://bliptext.com/articles/johannes-kepler) in 1609, though it was known to earlier [geometers](https://bliptext.com/articles/geometer).\nIt was depicted in [Pacioli](https://bliptext.com/articles/pacioli)'s [De Divina Proportione](https://bliptext.com/articles/divina-proportione), 1509.\nIt is the simplest of five regular [polyhedral compounds](https://bliptext.com/articles/polyhedral-compound), and the only regular [compound of two tetrahedra](https://bliptext.com/articles/compound-of-two-tetrahedra).\nIt is also the least dense of the regular polyhedral compounds, having a density of 2.\nIt can be seen as a 3D extension of the [hexagram](https://bliptext.com/articles/hexagram): the [hexagram](https://bliptext.com/articles/hexagram) is a two-dimensional shape formed from two overlapping equilateral triangles, [centrally symmetric](https://bliptext.com/articles/central-symmetry) to each other, and in the same way the stellated octahedron can be formed from two [centrally symmetric](https://bliptext.com/articles/central-symmetry) overlapping tetrahedra.\nThis can be generalized to any desired amount of higher dimensions; the four-dimensional equivalent construction is the [compound of two 5-cells](https://bliptext.com/articles/compound-of-two-5-cells).\nIt can also be seen as one of the stages in the construction of a 3D [Koch snowflake](https://bliptext.com/articles/koch-snowflake), a fractal shape formed by repeated attachment of smaller tetrahedra to each triangular face of a larger figure.\nThe first stage of the construction of the Koch Snowflake is a single central tetrahedron, and the second stage, formed by adding four smaller tetrahedra to the faces of the central tetrahedron, is the stellated octahedron.\n\n# Construction\nThe stellated octahedron can be constructed in several ways:\n\n# Related concepts\nA compound of two spherical tetrahedra can be constructed, as illustrated.\nThe two tetrahedra of the compound view of the stellated octahedron are \"desmic\", meaning that (when interpreted as a line in [projective space](https://bliptext.com/articles/projective-space)) each edge of one tetrahedron crosses two opposite edges of the other tetrahedron.\nOne of these two crossings is visible in the stellated octahedron; the other crossing occurs at a point at infinity of the projective space, where each edge of one tetrahedron crosses the parallel edge of the other tetrahedron.\nThese two tetrahedra can be completed to a [desmic system](https://bliptext.com/articles/desmic-system) of three tetrahedra, where the third tetrahedron has as its four vertices the three crossing points at infinity and the centroid of the two finite tetrahedra.\nThe same twelve tetrahedron vertices also form the points of [Reye's configuration](https://bliptext.com/articles/reye-s-configuration).\nThe [stella octangula numbers](https://bliptext.com/articles/stella-octangula-number) are [figurate numbers](https://bliptext.com/articles/figurate-number) that count the number of balls that can be arranged into the shape of a stellated octahedron.\nThey are\n\n# In popular culture\nThe stellated octahedron appears with several other polyhedra and polyhedral compounds in [M. C. Escher](https://bliptext.com/articles/m-c-escher)'s print \"[Stars](https://bliptext.com/articles/stars-m-c-escher)\", and provides the central form in Escher's [Double Planetoid](https://bliptext.com/articles/double-planetoid) (1949).\nThe obelisk in the center of the Plaza de Europa ([Zaragoza](https://bliptext.com/articles/zaragoza)) in [Zaragoza](https://bliptext.com/articles/zaragoza), [Spain](https://bliptext.com/articles/spain), is surrounded by twelve stellated octahedral lampposts, shaped to form a three-dimensional version of the [Flag of Europe](https://bliptext.com/articles/flag-of-europe).\nSome modern mystics have associated this shape with the \"merkaba\", which according to them is a \"counter-rotating energy field\" named from an ancient Egyptian word.\nHowever, the word \"merkaba\" is actually [Hebrew](https://bliptext.com/articles/hebrew-language), and more properly refers to a [chariot](https://bliptext.com/articles/chariot) in the visions of [Ezekiel](https://bliptext.com/articles/ezekiel).\nThe resemblance between this shape and the two-dimensional [star of David](https://bliptext.com/articles/star-of-david) has also been frequently noted.\nThe musical project \"Miracle Musical\" (often stylized in its original Japanese title ミラクルミュージカル, pronounced \"mirakuru myujikaru\" ), spearheaded by [Tally Hall](https://bliptext.com/articles/tally-hall) member Joe Hawley along with bandmate Ross Federman and honorary bandmate Bora Karaca, makes multiple references towards the stellated octahedron as the stella octangula.\nThe shape is shown on the main website of the project, as well as the merchandise store.\nThe third song on their first and only studio album, \"Hawaii: Part II\", \"Black Rainbows\" features a lyric sung by [Madi Diaz](https://bliptext.com/articles/madi-diaz) which simply says \"Stella octangula\".\n# [As a spherical tiling, the combined edges in the compound of two tetrahedra form a rhombic dodecahedron. | upload.wikimedia.org/wikipedia/commons/f/f7/Spherical///compound///of///two///tetrahedra.png]","revisionCount":3},"hasMore":false,"nextCursor":null}

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