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-The **Pseudorhombicuboctahedron** is a captivating [polyhedron](/wiki/polyhedron), a geometric marvel resembling the [rhombicuboctahedron](/wiki/rhombicuboctahedron) but with a subtle twist that breaks its uniform symmetry. This non-uniform variant, also known as the **elongated square gyrobicupola** (J<sub>37</sub>), is classified as a [Johnson Solid](/wiki/Johnson_Solid) because its faces are regular polygons, but it is not a [uniform polyhedron](/wiki/uniform_polyhedron) or an [Archimedean Solid](/wiki/Archimedean_Solid) due to its non-standard [vertex configuration](/wiki/vertex_configuration). Unlike uniform polyhedra, not all of its vertices are congruent, meaning the arrangement of faces around each vertex is not identical throughout the solid. This distinction is crucial for its classification as a Johnson Solid.
-Structurally, the pseudorhombicuboctahedron, or **elongated square gyrobicupola**, can be understood through its decomposition, clearly shown in the image. It is formed by taking a central [octagonal prism](/wiki/Octagonal_Prism) and attaching two [square cupolas](/wiki/Square_Cupola) to its opposite octagonal faces. Crucially, one of these cupolas is then rotated by 45 degrees relative to the other (a "gyro" operation), breaking the higher symmetry of the standard [rhombicuboctahedron](/wiki/rhombicuboctahedron). In a true rhombicuboctahedron, both cupolas would be perfectly aligned, resulting in greater symmetry. This 45-degree rotation is the defining characteristic that affects the arrangement of its faces and vertices, making it unique and distinguishing it from its uniform counterpart.
-Like the rhombicuboctahedron, it has 26 faces: 18 squares and 8 triangles. It also possesses 24 vertices and 48 edges. However, unlike the true rhombicuboctahedron, not all of its vertices are equivalent. Specifically, it has two types of vertices: those where three squares and one triangle meet, and those where two squares and two triangles meet. This lack of vertex transitivity is precisely why it falls outside the category of uniform polyhedra. Its [symmetry](/wiki/Symmetry) group is D<sub>4d</sub>, representing dihedral symmetry of order 4, which is significantly lower than the full octahedral symmetry (O<sub>h</sub>) of its uniform counterpart, further highlighting its distinct, lower symmetry arrangement.
-This unique construction highlights the diversity within [convex polyhedra](/wiki/convex_polyhedra) and serves as an excellent example of how slight modifications can lead to entirely new geometric classifications, challenging traditional definitions of regularity. The concept of "pseudo" polyhedra extends to other shapes as well, such as the [Pseudo Great Rhombicuboctahedron](/wiki/Pseudo_Great_Rhombicuboctahedron), which similarly introduces a break in uniform symmetry within a related family of polyhedra.
+The **Pseudorhombicuboctahedron** is a captivating [polyhedron](/wiki/polyhedron), a geometric marvel resembling the [rhombicuboctahedron](/wiki/rhombicuboctahedron) but with a subtle twist that breaks its uniform symmetry. This non-uniform variant, also known as the **elongated square gyrobicupola** (J<sub>37</sub>), is classified as a [Johnson Solid](/wiki/Johnson_Solid) because its faces are regular polygons, but it is not a [uniform polyhedron](/wiki/uniform_polyhedron) or an [Archimedean Solid](/wiki/Archimedean_Solid) due to its non-standard [vertex configuration](/wiki/vertex_configuration). Unlike uniform polyhedra, not all of its vertices are congruent; this means the arrangement of faces around each vertex is not identical throughout the solid, a property known as a lack of [vertex transitivity](/wiki/Vertex_Transitivity). This distinction is crucial for its classification as a Johnson Solid, which permits regular polygonal faces but does not require uniform vertices or edges.
+Structurally, the pseudorhombicuboctahedron, or **elongated square gyrobicupola**, can be understood through its decomposition, clearly shown in the image. It is formed by taking a central [octagonal prism](/wiki/Octagonal_Prism) and attaching two [square cupolas](/wiki/Square_Cupola) to its opposite octagonal faces. Crucially, one of these cupolas is then rotated by 45 degrees relative to the other (a "gyro" operation). This rotation directly breaks the higher symmetry of the standard [rhombicuboctahedron](/wiki/rhombicuboctahedron), where both cupolas would be perfectly aligned, leading to a much more symmetrical structure. The 45-degree rotational displacement of one cupola relative to the other is the defining characteristic that alters the arrangement of its faces and vertices, making it unique and distinguishing it from its uniform counterpart. This "gyro" operation results in a noticeable twist that is central to its pseudo-uniform nature.
+Like the rhombicuboctahedron, the pseudorhombicuboctahedron has 26 faces: 18 squares and 8 triangles. It also possesses 24 vertices and 48 edges. However, unlike the true rhombicuboctahedron, not all of its vertices are equivalent. Specifically, it has two types of vertices: 16 vertices where three squares and one triangle meet (often denoted as (3.4.4.4)), and 8 vertices where two squares and two triangles meet (denoted as (3.4.3.4)). This fundamental lack of [vertex transitivity](/wiki/Vertex_Transitivity) is precisely why it falls outside the category of uniform polyhedra and Archimedean solids. Its [symmetry group](/wiki/Symmetry_Group) is D<sub>4d</sub>, representing dihedral symmetry of order 4. This implies a single 4-fold rotation axis passing through the centers of the two octagonal faces, along with perpendicular 2-fold rotation axes and dihedral mirror planes. This is a significantly lower level of [symmetry](/wiki/Symmetry) compared to the full octahedral symmetry (O<sub>h</sub>) of its uniform counterpart, further highlighting its distinct, lower symmetry arrangement.
+This unique construction highlights the diversity within [convex polyhedra](/wiki/Convex_Polyhedra) and serves as an excellent example of how slight modifications, such as a rotational twist, can lead to entirely new geometric classifications. These "pseudo" forms challenge traditional definitions of regularity by demonstrating shapes that visually resemble their uniform counterparts but fundamentally differ in their underlying symmetry properties. The concept of "pseudo" polyhedra extends to other shapes as well, such as the [Pseudo Great Rhombicuboctahedron](/wiki/Pseudo_Great_Rhombicuboctahedron), which similarly introduces a break in uniform symmetry within a related family of polyhedra, showcasing a broader principle in geometric classification.
+- [Polyhedron](/wiki/Polyhedron)
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-The **Pseudorhombicuboctahedron** is a captivating [polyhedron](/wiki/polyhedron), a geometric marvel resembling the [rhombicuboctahedron](/wiki/rhombicuboctahedron) but with a subtle twist that breaks its uniform symmetry. This non-uniform variant, also known as the **elongated square gyrobicupola** (J<sub>37</sub>), is classified as a [Johnson Solid](/wiki/Johnson_Solid) because its faces are regular polygons, but it is not a [uniform polyhedron](/wiki/uniform_polyhedron) or an [Archimedean Solid](/wiki/Archimedean_Solid) due to its non-standard [vertex configuration](/wiki/vertex_configuration).
-Structurally, the pseudorhombicuboctahedron, or **elongated square gyrobicupola**, can be understood through its decomposition, clearly shown in the image. It is formed by taking a central [octagonal prism](/wiki/Octagonal_Prism) and attaching two [square cupolas](/wiki/Square_Cupola) to its opposite octagonal faces. One of these cupolas is then rotated by 45 degrees relative to the other (a "gyro" operation), breaking the higher symmetry of the standard [rhombicuboctahedron](/wiki/rhombicuboctahedron), which would have both cupolas aligned. This rotation affects the arrangement of its faces and vertices, making it unique.
-Like the rhombicuboctahedron, it has 26 faces: 18 squares and 8 triangles. It also possesses 24 vertices and 48 edges. However, unlike the true rhombicuboctahedron, not all of its vertices are equivalent, which is why it falls outside the category of uniform polyhedra. Its [symmetry](/wiki/Symmetry) group is D<sub>4d</sub>, which is lower than the full octahedral symmetry of its uniform counterpart, highlighting its distinct, lower symmetry arrangement.
-This unique construction highlights the diversity within [convex polyhedra](/wiki/convex_polyhedra) and serves as an excellent example of how slight modifications can lead to entirely new geometric classifications, challenging traditional definitions of regularity.
-- [Polyhedron](/wiki/Polyhedron)
+The **Pseudorhombicuboctahedron** is a captivating [polyhedron](/wiki/polyhedron), a geometric marvel resembling the [rhombicuboctahedron](/wiki/rhombicuboctahedron) but with a subtle twist that breaks its uniform symmetry. This non-uniform variant, also known as the **elongated square gyrobicupola** (J<sub>37</sub>), is classified as a [Johnson Solid](/wiki/Johnson_Solid) because its faces are regular polygons, but it is not a [uniform polyhedron](/wiki/uniform_polyhedron) or an [Archimedean Solid](/wiki/Archimedean_Solid) due to its non-standard [vertex configuration](/wiki/vertex_configuration). Unlike uniform polyhedra, not all of its vertices are congruent, meaning the arrangement of faces around each vertex is not identical throughout the solid. This distinction is crucial for its classification as a Johnson Solid.
+Structurally, the pseudorhombicuboctahedron, or **elongated square gyrobicupola**, can be understood through its decomposition, clearly shown in the image. It is formed by taking a central [octagonal prism](/wiki/Octagonal_Prism) and attaching two [square cupolas](/wiki/Square_Cupola) to its opposite octagonal faces. Crucially, one of these cupolas is then rotated by 45 degrees relative to the other (a "gyro" operation), breaking the higher symmetry of the standard [rhombicuboctahedron](/wiki/rhombicuboctahedron). In a true rhombicuboctahedron, both cupolas would be perfectly aligned, resulting in greater symmetry. This 45-degree rotation is the defining characteristic that affects the arrangement of its faces and vertices, making it unique and distinguishing it from its uniform counterpart.
+Like the rhombicuboctahedron, it has 26 faces: 18 squares and 8 triangles. It also possesses 24 vertices and 48 edges. However, unlike the true rhombicuboctahedron, not all of its vertices are equivalent. Specifically, it has two types of vertices: those where three squares and one triangle meet, and those where two squares and two triangles meet. This lack of vertex transitivity is precisely why it falls outside the category of uniform polyhedra. Its [symmetry](/wiki/Symmetry) group is D<sub>4d</sub>, representing dihedral symmetry of order 4, which is significantly lower than the full octahedral symmetry (O<sub>h</sub>) of its uniform counterpart, further highlighting its distinct, lower symmetry arrangement.
+This unique construction highlights the diversity within [convex polyhedra](/wiki/convex_polyhedra) and serves as an excellent example of how slight modifications can lead to entirely new geometric classifications, challenging traditional definitions of regularity. The concept of "pseudo" polyhedra extends to other shapes as well, such as the [Pseudo Great Rhombicuboctahedron](/wiki/Pseudo_Great_Rhombicuboctahedron), which similarly introduces a break in uniform symmetry within a related family of polyhedra.
+- [Rhombicuboctahedron](/wiki/Rhombicuboctahedron)
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-The **Pseudorhombicuboctahedron** is a captivating [polyhedron](/wiki/polyhedron), a geometric marvel resembling the [rhombicuboctahedron](/wiki/rhombicuboctahedron) but with a subtle twist that breaks its uniform symmetry. This non-uniform variant, also known as the **elongated square gyrobicupola** (J<sub>37</sub>), is classified as a [Johnson Solid](/wiki/Johnson_Solid) because its faces are regular polygons, but it is not a [uniform polyhedron](/wiki/uniform_polyhedron) or an [Archimedean Solid](/wiki/Archimedean_Solid) due to its non-standard vertex configuration.
-Structurally, it can be visualized as a rhombicuboctahedron where one of its square cupolas has been rotated by 45 degrees relative to the other. This rotation affects the arrangement of its faces and vertices, making it unique. Like the rhombicuboctahedron, it has 26 faces: 18 squares and 8 triangles. It also possesses 24 vertices and 48 edges. However, unlike the true [rhombicuboctahedron](/wiki/rhombicuboctahedron), not all of its vertices are equivalent, which is why it falls outside the category of uniform polyhedra. Its [symmetry](/wiki/Symmetry) group is D<sub>4d</sub>, which is lower than the full octahedral symmetry of its uniform counterpart.
+The **Pseudorhombicuboctahedron** is a captivating [polyhedron](/wiki/polyhedron), a geometric marvel resembling the [rhombicuboctahedron](/wiki/rhombicuboctahedron) but with a subtle twist that breaks its uniform symmetry. This non-uniform variant, also known as the **elongated square gyrobicupola** (J<sub>37</sub>), is classified as a [Johnson Solid](/wiki/Johnson_Solid) because its faces are regular polygons, but it is not a [uniform polyhedron](/wiki/uniform_polyhedron) or an [Archimedean Solid](/wiki/Archimedean_Solid) due to its non-standard [vertex configuration](/wiki/vertex_configuration).
+
+*The image illustrates the construction of the pseudorhombicuboctahedron. From left to right: the elongated square gyrobicupola, its decomposition into two square cupolas and an octagonal prism, and the rhombicuboctahedron.*
+Structurally, the pseudorhombicuboctahedron, or **elongated square gyrobicupola**, can be understood through its decomposition, clearly shown in the image. It is formed by taking a central [octagonal prism](/wiki/Octagonal_Prism) and attaching two [square cupolas](/wiki/Square_Cupola) to its opposite octagonal faces. One of these cupolas is then rotated by 45 degrees relative to the other (a "gyro" operation), breaking the higher symmetry of the standard [rhombicuboctahedron](/wiki/rhombicuboctahedron), which would have both cupolas aligned. This rotation affects the arrangement of its faces and vertices, making it unique.
+Like the rhombicuboctahedron, it has 26 faces: 18 squares and 8 triangles. It also possesses 24 vertices and 48 edges. However, unlike the true rhombicuboctahedron, not all of its vertices are equivalent, which is why it falls outside the category of uniform polyhedra. Its [symmetry](/wiki/Symmetry) group is D<sub>4d</sub>, which is lower than the full octahedral symmetry of its uniform counterpart, highlighting its distinct, lower symmetry arrangement.
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-The **Pseudorhombicuboctahedron** is a captivating [polyhedron](/wiki/polyhedron), a geometric marvel resembling the [rhombicuboctahedron](/wiki/rhombicuboctahedron) but with a subtle twist. This non-uniform variant often arises from a 45-degree rotation of one of its caps, creating a shape that beautifully defies standard classification among the true uniform solids.
-- [Geometric Solid](/wiki/Geometric_Solid)
+The **Pseudorhombicuboctahedron** is a captivating [polyhedron](/wiki/polyhedron), a geometric marvel resembling the [rhombicuboctahedron](/wiki/rhombicuboctahedron) but with a subtle twist that breaks its uniform symmetry. This non-uniform variant, also known as the **elongated square gyrobicupola** (J<sub>37</sub>), is classified as a [Johnson Solid](/wiki/Johnson_Solid) because its faces are regular polygons, but it is not a [uniform polyhedron](/wiki/uniform_polyhedron) or an [Archimedean Solid](/wiki/Archimedean_Solid) due to its non-standard vertex configuration.
+Structurally, it can be visualized as a rhombicuboctahedron where one of its square cupolas has been rotated by 45 degrees relative to the other. This rotation affects the arrangement of its faces and vertices, making it unique. Like the rhombicuboctahedron, it has 26 faces: 18 squares and 8 triangles. It also possesses 24 vertices and 48 edges. However, unlike the true [rhombicuboctahedron](/wiki/rhombicuboctahedron), not all of its vertices are equivalent, which is why it falls outside the category of uniform polyhedra. Its [symmetry](/wiki/Symmetry) group is D<sub>4d</sub>, which is lower than the full octahedral symmetry of its uniform counterpart.
+This unique construction highlights the diversity within [convex polyhedra](/wiki/convex_polyhedra) and serves as an excellent example of how slight modifications can lead to entirely new geometric classifications, challenging traditional definitions of regularity.
+- [Uniform Polyhedron](/wiki/Uniform_Polyhedron)
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+The **Pseudorhombicuboctahedron** is a captivating [polyhedron](/wiki/polyhedron), a geometric marvel resembling the [rhombicuboctahedron](/wiki/rhombicuboctahedron) but with a subtle twist. This non-uniform variant often arises from a 45-degree rotation of one of its caps, creating a shape that beautifully defies standard classification among the true uniform solids.
+## See also
+- [Archimedean Solid](/wiki/Archimedean_Solid)
+- [Johnson Solid](/wiki/Johnson_Solid)
+- [Geometric Solid](/wiki/Geometric_Solid)
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