Net (polyhedron)

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A net of a regular dodecahedron Dodecahedron flat.svg
A net of a regular dodecahedron
The eleven nets of a cube The 11 cubic nets.svg
The eleven nets of a cube

In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard. [1]

Contents

An early instance of polyhedral nets appears in the works of Albrecht Dürer, whose 1525 book A Course in the Art of Measurement with Compass and Ruler (Unterweysung der Messung mit dem Zyrkel und Rychtscheyd ) included nets for the Platonic solids and several of the Archimedean solids. [2] [3] These constructions were first called nets in 1543 by Augustin Hirschvogel. [4]

Existence and uniqueness

Four hexagons that, when glued to form a regular octahedron as depicted, produce folds across three of the diagonals of each hexagon. The edges between the hexagons remain unfolded. 4-hex octahedron.svg
Four hexagons that, when glued to form a regular octahedron as depicted, produce folds across three of the diagonals of each hexagon. The edges between the hexagons remain unfolded.

Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated. The edges that are cut from a convex polyhedron to form a net must form a spanning tree of the polyhedron, but cutting some spanning trees may cause the polyhedron to self-overlap when unfolded, rather than forming a net. [5] Conversely, a given net may fold into more than one different convex polyhedron, depending on the angles at which its edges are folded and the choice of which edges to glue together. [6] If a net is given together with a pattern for gluing its edges together, such that each vertex of the resulting shape has positive angular defect and such that the sum of these defects is exactly 4π, then there necessarily exists exactly one polyhedron that can be folded from it; this is Alexandrov's uniqueness theorem. However, the polyhedron formed in this way may have different faces than the ones specified as part of the net: some of the net polygons may have folds across them, and some of the edges between net polygons may remain unfolded. Additionally, the same net may have multiple valid gluing patterns, leading to different folded polyhedra. [7]

Unsolved problem in mathematics:
Does every convex polyhedron have a simple edge unfolding?

In 1975, G. C. Shephard asked whether every convex polyhedron has at least one net, or simple edge-unfolding. [8] This question, which is also known as Dürer's conjecture, or Dürer's unfolding problem, remains unanswered. [9] [10] [11] There exist non-convex polyhedra that do not have nets, and it is possible to subdivide the faces of every convex polyhedron (for instance along a cut locus) so that the set of subdivided faces has a net. [5] In 2014 Mohammad Ghomi showed that every convex polyhedron admits a net after an affine transformation. [12] Furthermore, in 2019 Barvinok and Ghomi showed that a generalization of Dürer's conjecture fails for pseudo edges, [13] i.e., a network of geodesics which connect vertices of the polyhedron and form a graph with convex faces.

Blooming a regular dodecahedron Net of dodecahedron.gif
Blooming a regular dodecahedron

A related open question asks whether every net of a convex polyhedron has a blooming, a continuous non-self-intersecting motion from its flat to its folded state that keeps each face flat throughout the motion. [14]

Shortest path

The shortest path over the surface between two points on the surface of a polyhedron corresponds to a straight line on a suitable net for the subset of faces touched by the path. The net has to be such that the straight line is fully within it, and one may have to consider several nets to see which gives the shortest path. For example, in the case of a cube, if the points are on adjacent faces one candidate for the shortest path is the path crossing the common edge; the shortest path of this kind is found using a net where the two faces are also adjacent. Other candidates for the shortest path are through the surface of a third face adjacent to both (of which there are two), and corresponding nets can be used to find the shortest path in each category. [15]

The spider and the fly problem is a recreational mathematics puzzle which involves finding the shortest path between two points on a cuboid.

Higher-dimensional polytope nets

The Dali cross, one of the 261 nets of the tesseract Tesseract2.svg
The Dalí cross, one of the 261 nets of the tesseract

A net of a 4-polytope, a four-dimensional polytope, is composed of polyhedral cells that are connected by their faces and all occupy the same three-dimensional space, just as the polygon faces of a net of a polyhedron are connected by their edges and all occupy the same plane. The net of the tesseract, the four-dimensional hypercube, is used prominently in a painting by Salvador Dalí, Crucifixion (Corpus Hypercubus) (1954). [16] The same tesseract net is central to the plot of the short story "—And He Built a Crooked House—" by Robert A. Heinlein. [17]

The number of combinatorially distinct nets of -dimensional hypercubes can be found by representing these nets as a tree on nodes describing the pattern by which pairs of faces of the hypercube are glued together to form a net, together with a perfect matching on the complement graph of the tree describing the pairs of faces that are opposite each other on the folded hypercube. Using this representation, the number of different unfoldings for hypercubes of dimensions 2, 3, 4, ... have been counted as

1, 11, 261, 9694, 502110, 33064966, 2642657228, ... (sequence A091159 in the OEIS )

See also

Related Research Articles

<span class="mw-page-title-main">Cube</span> Solid object with six equal square faces

In geometry, a cube is a three-dimensional solid object bounded by six square faces. It has twelve edges and eight vertices. It can be represented as a rectangular cuboid with six square faces, or a parallelepiped with equal edges. It is an example of many type of solids: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron.

<span class="mw-page-title-main">Dual polyhedron</span> Polyhedron associated with another by swapping vertices for faces

In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

<span class="mw-page-title-main">Polyhedron</span> 3D shape with flat faces, straight edges and sharp corners

In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.

<span class="mw-page-title-main">4-polytope</span> Four-dimensional geometric object with flat sides

In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.

In geometry, a cuboid is a quadrilateral-faced convex hexahedron, a polyhedron with six faces.

<span class="mw-page-title-main">Truncated octahedron</span> Archimedean solid

In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces, 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate 3-dimensional space, as a permutohedron.

<span class="mw-page-title-main">Regular polytope</span> Polytope with highest degree of symmetry

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. In particular, all its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension jn.

Cauchy's theorem is a theorem in geometry, named after Augustin Cauchy. It states that convex polytopes in three dimensions with congruent corresponding faces must be congruent to each other. That is, any polyhedral net formed by unfolding the faces of the polyhedron onto a flat surface, together with gluing instructions describing which faces should be connected to each other, uniquely determines the shape of the original polyhedron. For instance, if six squares are connected in the pattern of a cube, then they must form a cube: there is no convex polyhedron with six square faces connected in the same way that does not have the same shape.

<span class="mw-page-title-main">Flexible polyhedron</span>

In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex.

<span class="mw-page-title-main">Cut locus</span>

In differential geometry, the cut locus of a point p on a manifold is the closure of the set of all other points on the manifold that are connected to p by two or more distinct shortest geodesics. More generally, the cut locus of a closed set X on the manifold is the closure of the set of all other points on the manifold connected to X by two or more distinct shortest geodesics.

Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes.

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The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each other also have distinct metric spaces of surface distances, and it characterizes the metric spaces that come from the surface distances on polyhedra. It is named after Soviet mathematician Aleksandr Danilovich Aleksandrov, who published it in the 1940s.

In geometry, more specifically in polytope theory, Kalai's 3d conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989. It states that every d-dimensional centrally symmetric polytope has at least 3d nonempty faces.

Geometric Folding Algorithms: Linkages, Origami, Polyhedra is a monograph on the mathematics and computational geometry of mechanical linkages, paper folding, and polyhedral nets, by Erik Demaine and Joseph O'Rourke. It was published in 2007 by Cambridge University Press (ISBN 978-0-521-85757-4). A Japanese-language translation by Ryuhei Uehara was published in 2009 by the Modern Science Company (ISBN 978-4-7649-0377-7).

In computational geometry, the star unfolding of a convex polyhedron is a net obtained by cutting the polyhedron along geodesics through its faces. It has also been called the inward layout of the polyhedron, or the Alexandrov unfolding after Aleksandr Danilovich Aleksandrov, who first considered it.

Convex Polytopes is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and G. C. Shephard, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics.

<span class="mw-page-title-main">Blooming (geometry)</span>

In the geometry of convex polyhedra, blooming or continuous blooming is a continuous three-dimensional motion of the surface of the polyhedron, cut to form a polyhedral net, from the polyhedron into a flat and non-self-overlapping placement of the net in a plane. As in rigid origami, the polygons of the net must remain individually flat throughout the motion, and are not allowed to intersect or cross through each other. A blooming, reversed to go from the flat net to a polyhedron, can be thought of intuitively as a way to fold the polyhedron from a paper net without bending the paper except at its designated creases.

In computational geometry, the source unfolding of a convex polyhedron is a net obtained by cutting the polyhedron along the cut locus of a point on the surface of the polyhedron. The cut locus of a point consists of all points on the surface that have two or more shortest geodesics to . For every convex polyhedron, and every choice of the point on its surface, cutting the polyhedron on the cut locus will produce a result that can be unfolded into a flat plane, producing the source unfolding. The resulting net may, however, cut across some of the faces of the polyhedron rather than only cutting along its edges.

<span class="mw-page-title-main">Common net</span> Edge-joined polygon with multiple principle shapes

In geometry, a common net is a net that can be folded onto several polyhedra. To be a valid common net, there shouldn't exist any non-overlapping sides and the resulting polyhedra must be connected through faces. The research of examples of this particular nets dates back to the end of the 20th century, despite that, not many examples have been found. Two classes, however, have been deeply explored, regular polyhedra and cuboids. The search of common nets is usually made by either extensive search or the overlapping of nets that tile the plane.

References

  1. Wenninger, Magnus J. (1971), Polyhedron Models, Cambridge University Press
  2. Dürer, Albrecht (1525), Unterweysung der Messung mit dem Zyrkel und Rychtscheyd, Nürnberg: München, Süddeutsche Monatsheft, pp. 139–152. English translation with commentary in Strauss, Walter L. (1977), The Painter's Manual, New York{{citation}}: CS1 maint: location missing publisher (link)
  3. Schreiber, Fischer, and Sternath claim that, earlier than Dürer, Leonardo da Vinci drew several nets for Luca Pacioli's Divina proportione , including a net for the regular dodecahedron. However, these cannot be found in online copies of the 1509 first printed edition of this work nor in the 1498 Geneva ms 210, so this claim should be regarded as unverified. See: Schreiber, Peter; Fischer, Gisela; Sternath, Maria Luise (July 2008), "New light on the rediscovery of the Archimedean solids during the Renaissance", Archive for History of Exact Sciences, 62 (4): 457–467, doi:10.1007/s00407-008-0024-z, JSTOR   41134285
  4. Friedman, Michael (2018), A History of Folding in Mathematics: Mathematizing the Margins, Science Networks. Historical Studies, vol. 59, Birkhäuser, p. 8, doi:10.1007/978-3-319-72487-4, ISBN   978-3-319-72486-7
  5. 1 2 Demaine, Erik D.; O'Rourke, Joseph (2007), "Chapter 22. Edge Unfolding of Polyhedra", Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, pp. 306–338
  6. Malkevitch, Joseph, "Nets: A Tool for Representing Polyhedra in Two Dimensions", Feature Columns, American Mathematical Society , retrieved 2014-05-14
  7. Demaine, Erik D.; Demaine, Martin L.; Lubiw, Anna; O'Rourke, Joseph (2002), "Enumerating foldings and unfoldings between polygons and polytopes", Graphs and Combinatorics , 18 (1): 93–104, arXiv: cs.CG/0107024 , doi:10.1007/s003730200005, MR   1892436, S2CID   1489
  8. Shephard, G. C. (1975), "Convex polytopes with convex nets", Mathematical Proceedings of the Cambridge Philosophical Society, 78 (3): 389–403, Bibcode:1975MPCPS..78..389S, doi:10.1017/s0305004100051860, MR   0390915, S2CID   122287769
  9. Weisstein, Eric W., "Shephard's Conjecture", MathWorld
  10. Moskovich, D. (June 4, 2012), "Dürer's conjecture", Open Problem Garden
  11. Ghomi, Mohammad (2018-01-01), "Dürer's Unfolding Problem for Convex Polyhedra", Notices of the American Mathematical Society, 65 (1): 25–27, doi: 10.1090/noti1609
  12. Ghomi, Mohammad (2014), "Affine unfoldings of convex polyhedra", Geom. Topol. , 18 (5): 3055–3090, arXiv: 1305.3231 , Bibcode:2013arXiv1305.3231G, doi:10.2140/gt.2014.18.3055, S2CID   16827957
  13. Barvinok, Nicholas; Ghomi, Mohammad (2019-04-03), "Pseudo-Edge Unfoldings of Convex Polyhedra", Discrete & Computational Geometry , 64 (3): 671–689, arXiv: 1709.04944 , doi:10.1007/s00454-019-00082-1, ISSN   0179-5376, S2CID   37547025
  14. Miller, Ezra; Pak, Igor (2008), "Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings", Discrete & Computational Geometry , 39 (1–3): 339–388, doi: 10.1007/s00454-008-9052-3 , MR   2383765
  15. O’Rourke, Joseph (2011), How to Fold It: The Mathematics of Linkages, Origami and Polyhedra, Cambridge University Press, pp. 115–116, ISBN   9781139498548
  16. Kemp, Martin (1 January 1998), "Dali's dimensions", Nature , 391 (6662): 27, Bibcode:1998Natur.391...27K, doi: 10.1038/34063 , S2CID   5317132
  17. Henderson, Linda Dalrymple (November 2014), "Science Fiction, Art, and the Fourth Dimension", in Emmer, Michele (ed.), Imagine Math 3: Between Culture and Mathematics, Springer International Publishing, pp. 69–84, doi:10.1007/978-3-319-01231-5_7, ISBN   978-3-319-01230-8