{"id":1034,"date":"2021-06-20T11:40:13","date_gmt":"2021-06-20T11:40:13","guid":{"rendered":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/?page_id=1034"},"modified":"2022-01-24T12:03:34","modified_gmt":"2022-01-24T12:03:34","slug":"https-en-wikipedia-org-wiki-honeycomb_geometry","status":"publish","type":"page","link":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/https-en-wikipedia-org-wiki-honeycomb_geometry\/","title":{"rendered":"Space Filling Cubic Polyhedra."},"content":{"rendered":"\n

The cubic honeycomb is the only regular honeycomb<\/a> in Euclidian 3-space. There are only 5 space-filling polyhedra using translations only, called parallelohedra. Three of them have cubic symmetry. They are the Wigner-Seitz cells of the primitive, body-centered and face-centered cubic lattices (see fig. 1)<\/p>\n\n\n\n

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Fig.1 Wigner Seitz cells of the body-centered<\/em>, face-centered and primitivecubic honeycombs. In the lower half of the picture the corresponding basic polyhedra are shown from which the WS-cells may be obtained by applying all possible cubic mirror operations. <\/figcaption><\/figure>\n\n\n\n

The cube can be divided into 48 tetrahedra by the mirror planes of the cubic system. It is in fact a first type Hill tetrahedron<\/a> <\/p>\n\n\n\n

The basic tetrahedron of the primitive cube can be divided into two identical halves for the body-centered cubic Wigner-Seitz cell, or truncated octahedron, and into four identical basic polyhedra for the face-centered cubic Wigner-Seitz cell, the rhombic dodecahedron.<\/p>\n\n\n\n

The truncated octahedron is not the only space filling polyhedron on a body centered cubic lattice, because the basic tetrahedron of the primitive cube can be divided into identical halves in many different ways by a separation plane containing the twofold axis connecting the two halves (fig.2). Two of the polyhedra obtained in this way are the Hill tetrahedra type 2 and 3.<\/p>\n\n\n\n

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Fig. 2 Left: Plane disecting the basic cubic tetrahedron containing a two-fold axis.
Right: By applying all possible cubic mirror planes an infinite polytopic surface is obtained, separating two congruent infinite polyhedra. <\/figcaption><\/figure>\n\n\n\n

All examples of the bcc spacefilling polyhedra shown in fig.3 are non-convex in contrast to the truncated octahedron, the WS cell. <\/p>\n\n\n\n

The magenta object is the well-known stellated rhombic dodecahedron, also known as the Escher solid. The basic tetrahedron is a combination of two basic tetrahedra of the fcc Wigner Seitz cell . The separation plane contains the face center of the cube. Its basic tetrahedron is a Hill type 2 tetrahedron. <\/p>\n\n\n\n

The grey object is a cubic cross. The separation plane contains a point halfway the cube edge and a face center. <\/p>\n\n\n\n

The orange object is special, because the separation plane contains the cube center. Its basic tetrahedron is a Hill type 3 tetrahedron. The honeycomb can alternatively be built by the space filling star-shaped hexadecahedron ( in projection looking like a wind rose) shown in the upper left part of the picture. In the space filling tiling six of these stars have a body diagonal of the cube in common. These stars fit into the slots at the cube edges. In fig. 4 resin prints of the wind-rose-like objects are shown.<\/p>\n\n\n\n

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Fig. 3 Space filling polyhedra on a body centered cubic lattice..<\/figcaption><\/figure>\n\n\n\n

The dividing surface does not have to be planar as long as it is also twofold rotation symmetric in the the way described previously. A famous example is the Schwartz primitive minimal surface<\/a> . It looks somewhat like the yellow polyhedron, but the cross section with the cube surface is a circle instead of an octagon.<\/p>\n\n\n\n

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Fig. 4. 3d resin prints of Wind Rose polyhedra. <\/figcaption><\/figure>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

The cubic honeycomb is the only regular honeycomb in Euclidian 3-space. There are only 5 space-filling polyhedra using translations only, called parallelohedra. Three of them have cubic symmetry. They are the Wigner-Seitz cells of the primitive, body-centered and face-centered cubic lattices (see fig. 1) The cube can be divided into 48 tetrahedra by the mirror … Continue reading Space Filling Cubic Polyhedra.<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1034","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/pages\/1034","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/comments?post=1034"}],"version-history":[{"count":23,"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/pages\/1034\/revisions"}],"predecessor-version":[{"id":1126,"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/pages\/1034\/revisions\/1126"}],"wp:attachment":[{"href":"https:\/\/www.aperiodictiling.org\/wpaperiodictiling\/index.php\/wp-json\/wp\/v2\/media?parent=1034"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}