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Topic: Vertex figures

Related Topics

vertex figures

Vertex shaders

Vertex cover problem

vertex algebra

proper vertex coloring

vertex graph theory

Vertex Cover

Vertex transitive graph

vertex angle

vertex coloring

matched vertex

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	Figures and polygons
		A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others.
		A figure is convex if every line segment drawn between any two points inside the figure lies entirely inside the figure.
		A figure that is not convex is called a concave figure.
	www.mathleague.com /help/geometry/polygons.htm (620 words)

	Vertex figure - Wikipedia, the free encyclopedia
		The vertex figure of a triangular prism is an isosceles triangle.
		A short-hand notation for this vertex figure is 3.4.4
		The vertex figure of a truncated cubic honeycomb is a nonuniform square pyramid.
	en.wikipedia.org /wiki/Vertex_figure (574 words)

	PolyGloss
		The gosset figure with a vertex with simplex-vertex symmetry.
		The vertex figure is o3m4m3o, the convex hull of dual 24-chora.
		The surtope of a dual that corresponds to the surtope of the figure.
	www.geocities.com /os2fan2/gloss.htm (16747 words)

	Mean Distance from Vertex to Interior of Plane Figures
		To determine the mean distance from a vertex to the interior points of a given triangle, we can begin by noting the expression for the position of the centroid of an arbitrary triangle relative to one vertex.
		The mean distance from the vertex to the points of the triangle is given by the average of D(t) for t ranging from 0 to 1.
		Naturally the mean distance (from the appropriate vertex) of the points of a right triangle is the same as the mean distance for the isosceles triangle constructed by placing two of the right triangles together, "back-to-back".
	www.mathpages.com /home/kmath283/kmath283.htm (977 words)

	Vertex figures
		Flat vertex figures have some advantages, not least that they are easier to draw on a sheet of paper, and stand some chance of behaving properly when reciprocated with respect to a circle or sphere.
		Figure 5 shows the more commonly studied one, in which the vertex P (pole) is outside the sphere and the plane of the dual face (polar) intersects the sphere.
		For an intrinsic vertex figure we must ask which region of space around the vertex is interior to the polyhedron (or if this is not evident, as it need not be, which regions of the planes between edge lines are boundaries of the polyhedron).
	www.steelpillow.com /polyhedra/vertex_figures/VertexFigures.htm (0 words)

	Polytopes, duality and precursors
		A vertex figure of a polyhedron may loosely be thought of as the polygonal surface revealed when a corner of the polyhedron is sliced off.
		A polar vertex figure occurs as the reciprocal figure to the face of the reciprocal polyhedron; its geometry depends also on that of the associated reciprocating sphere.
		A corner figure is a side in the vertex figure, and an edge section is a corner in the figure.
	www.steelpillow.com /polyhedra/generators/generators.htm (0 words)

	[No title] (Site not responding. Last check: )
		It was only a month or two ago that I re-listed all the vertex-figures of the archimedean 4-polytopes (not that I have the list here now, of course), and it's easy to see that not too many of them include regular skew polygons among the edges of their vertex figures.
		I see the English of that got a bit mixed up, since "vertex figures" was already the subject of the subordinate clause, but you see what I mean anyway.
		I should imagine that a better way to find examples would be to first find the maps in abstracto, then do the eigenvalue-type graph theory to find their embeddings, which might be in quite high-dimensional spaces.
	www.ics.uci.edu /~eppstein/junkyard/archimedean.html (270 words)

	Uniform Polyhedra
		In a uniform polyhedron, every face is required to be a regular polygon, and every vertex is required to be identical, but the faces need not be identical.
		The duals to the uniform polyhedra are facially regular, meaning that they are composed of a single type of face, every face in the same relation to the whole, and their vertex figures are regular polygons.
		There is also a kind of crossed antiprism, e.g., the pentagrammatic crossed antiprism also (5/2, 3, 3, 3), but the vertex figure is a butterfly-shaped (crossed) quadrilateral and the triangular faces cross the n-fold axis.
	www.georgehart.com /virtual-polyhedra/uniform-info.html (0 words)

	Consistent Vertex Descriptions
		All the faces are retrograde here (the fractions are greater than 1/2), which doesn't make the faces wrap around the vertex twice as is required, but rather makes them wrap backwards around the vertex (but still only once).
		If you look at any model that has both non-retrograde and retrograde triangles in its vertex description (eg the tetrahemihexahedron, "3, 4, 3/2, 4"), you will see that they both mark out the same angle around the vertex, but in opposite directions.
		Since this is the only program to use the vertex description alone in its calculations, they must be mathematically consistent with each other.
	home.aanet.com.au /robertw/VertexDesc.html (625 words)

	N-D Geometrical Figures
		The main trick in computing the figures is using permutations to generate the vertex sets and then post-analysis of the vertex sets to extract other features.
		The "names" {p,q} are the Schlafli symbols for the five regular figures in 3-D and {p,q,r} are the same for the six regular figures in 4-D. Vertex Sets
		There are many related kinds of figures (concave, expansions, prisms) in the Mathematica pages but they are not treated here.
	www.cs.dartmouth.edu /~mckeeman/references/polytopes/polydemo.html (1397 words)

	vertex figure - Information from Reference.com
		The vertex figure of a triangular prism is an isosceles triangle.
		This vertex figure has a 3-dimensional structure since the faces are not in...
		Tes - tesseract - cells are 8 cubes, its vertex figure is a tet,...
	dictionary.reference.com /search?q=vertex%20figure (322 words)

	Body
		The Geometer's Sketchpad has the potential to change the teaching and learning of geometry from the memorization of definitions and proofs to a subject that is meaningful and dynamic.
		Figures can be created using the Euclidean tools e.g., compass and straight edge.
		These figures can then be transformed, while preserving the geometric relationships of the constructions.
	math.nsu.edu /math/courses/500l/skpdtk.htm (1135 words)

	Four Dimensional Figures Page
		The only other names for such a figure that I had seen in the literature, “polyhedroid” and “hypersolid,” seem uninspired and inappropriate, because they’re too close to terms for three-dimensional polytopes; the ending -oid connotes similarity or resemblance; and the prefix hyper- is badly overused.
		The numerals on the edges in the pictures denote the number of sides of the corresponding regular-polygonal faces at a vertex of the polychoron; the length of an edge labeled p is thus 2cos(pi/p).
		Vertex figures, however, are not regular or uniform unless specifically denoted as such; their size assumes an Archimedean polytope of unit edge length.
	members.aol.com /Polycell/uniform.html (0 words)

	Math Forum: Ask Dr. Math FAQ: Defining Geometric Figures
		The name of the figure is also commonly used for this region, and the area of the region is commonly called the area of the figure.
		A three-dimensional figure, sometimes called a solid figure, is a set of plane regions and surface regions, all lying in three-dimensional space.
		The name of the figure is also commonly used for this region, and the volume of the region is commonly called the volume of the figure.
	mathforum.org /dr.math/faq/formulas/faq.figuredef.html (622 words)

	Four Dimensional Figures Page (Site not responding. Last check: )
		The only other names for such a figure that I had seen in the literature, “polyhedroid” and “hypersolid,” seem uninspired and inappropriate, because they’re too close to terms for three-dimensional polytopes; the ending -oid connotes similarity or resemblance; and the prefix hyper- is badly overused.
		The prism section, Section 6, pictures 36 vertex figures and takes more than twice as long to download as Sections 1–5; Section 7, all of whose polychora are duplicates of others in the table, needs no illustrations.
		The numerals on the edges in the pictures denote the number of sides of the corresponding regular-polygonal faces at a vertex of the polychoron; the length of an edge labeled p is thus 2cos(pi/p).
	hometown.aol.com /Polycell/uniform.html (4238 words)

	Augmenting the great rhombicuboctahedron
		in the vertex figure is replaced by a (4,4) The original vertex figures become (4,4,4,4,4)/2 with new vertices of (4,4,3).
		This leaves the original vertex figures unchanged at (4,4,4,4,4)/2 but the new vertices of (4,4,3) become (4,4,3,3) vertices and further vertices are added of the form (3,3,3).
		All vertices are on the exterior of the figure.
	web.ukonline.co.uk /polyhedra/uniform/augmented/22.html (486 words)

	Intersections of elementary surfaces (Site not responding. Last check: )
		In the case of a partial intersection (one closed figure), on one of the surfaces there are lines not intersecting the other surface.
		Vertex line v and its traces are determined in those coordinate planes, in which basic polygons of the intersection surfaces are located.
		Ideal vertex line is located in the side plane, auxhiliary vertex planes are parallel to the side plane.
	www.km.sjf.stuba.sk /Geometria/PREDNASKY/eleminter.htm (1619 words)

	[No title]
		This joins all of the rough sides of the slab figures, and we're left with a figure bounded by unbounded flat {8,3}'s, and detail completey inside, representing the verticies, edges and planes of a {8,3,4}.
		Of the cuboctahedron, the icosadodecahedron and rhombi-icosadodecahedron, the vertix figure of these are reptangles where the opposite sides are of the same nature, different to the other pair.
		Of the rhombicuboctahedron, the vertix figure is a reptangle, with three sides becominging squares, and the fourth a triangle.
	www.ics.uci.edu /~eppstein/junkyard/buckyball.html (3314 words)

	Vertex - production control comes of age - Support: Answers from the Workshop (Site not responding. Last check: )
		But VERTEX is not just a wordprocessor or spreadsheet or relational database where it is easy to spot and manually change wrong or incomplete data.
		The ability to report on these figures through a stock system tailored to your industry is a significant advantage when compared to the accounting system "going it alone" on a more general basis.
		So if you are thinking of upgrading your hardware or have had problems getting Vertex ticket printing routines to recognise the "dongle" with Windows 2000 workstations, you should contact us for the latest download.
	www.vertexplus.co.uk /work.htm (2172 words)

	Uniform Polyhedra
		Allowing for non-convex faces and vertex figures, there are 75 such polyhedra, as well as 2 infinite families of prisms and antiprisms.
		A recently discovered uniform way of computing their vertex coordinates [Harel93] is the basis for a program to display all of these solids, among which are many beautiful and stunning shapes.
		The vertex configuration is the sequence of faces arranged around a vertex.
	www.mathconsult.ch /showroom/unipoly/unipoly.html (0 words)

	The Mathematica Programmer: Uniform Polyhedra -- from Wolfram Library Archive
		Allowing for non-convex faces and vertex figures, there are 75 such polyhedra, as well as two infinite families of prisms and antiprisms.
		A recently discovered uniform way of computing their vertex coordinates is the basis for our program to display all of these solids, among which are many beautiful and stunning shapes.
		We will also develop some utilities for drawing spherical triangles and for storing properties of objects in a flexible way.
	library.wolfram.com /infocenter/Articles/2254 (103 words)

	Three Dimensional Figures
		See the figure at the right for a typical 2-D representation and chapter 1 for a perspective drawing.
		In a cone, the lateral edge is any segment whose endpoints are the vertex and a point on the base circle.
		A plane section of a three-dimensional figure is the intersection of that figure with a plane.
	www.andrews.edu /~calkins/math/webtexts/geom09.htm (2729 words)

	Beyond3D Forum - How many Polys can the New Consoles Push? (Site not responding. Last check: )
		Triangle: The plane figure formed by connecting three points not in a straight line by straight line segments; a three-sided polygon.
		Vertex: The point at which the sides of an angle intersect.
		For that reason I actually think quoting vertex figures is more accurate - there's less ambiguity about what a vertex is. But I've no doubt that when you see a company quote x triangles per sec, that means x vertices per sec.
	www.beyond3d.com /forum/printthread.php?t=22233 (2157 words)

	Tiling #5
		The vertex figure of a tiling is the pattern of polygons surrounding a vertex point.
		Use the table in #16 to determine all of the possible vertex figures of regular polygons with at most 12 sides.
		In addition to the vertex figures of #19, exactly five others are possible, each involving one polygon with more than 12 sides.
	home.snu.edu /~lturner/MC-GeoTop/Tiling5.htm (91 words)

	Space Figures (Site not responding. Last check: )
		Many of them have various numbers of faces, which is a side of the space figure, vertex or vertices (plural), edges which is where two faces of the space figure meet and vertex or vertices (plural) which are the corners of the various space figures.
		A rectangular prism is a space figure with six faces, eight vertices, and opposite sides parallel.
		A sphere is a space figure in which each point on the outside is equidistant from the center of the sphere.
	www.mcwdn.org /Geometry/SpaceFigures.html (128 words)

	Math Forum: What Is a Tessellation?
		Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees.
		For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane.
		Now keep going around the vertex in either direction, finding the number of sides of the polygons until you get back to the polygon you started with.
	www.mathforum.org /sum95/suzanne/whattess.html (753 words)

	Relations and sizes - Congruent figures - In Depth (Site not responding. Last check: )
		If two figures are congruent, then their corresponding parts are congruent.
		Congruent figures are named in the order of their corresponding parts.
		The side with one hash mark in triangle ABC corresponds to the side with one hash mark in triangle DFE.
	www.math.com /school/subject3/lessons/S3U3L1DP.html (251 words)

	Uniform Polychora
		- this includes those with vertex figures shaped like tetrahedra with bilateral symmetry.
		- this includes those with trapezoid pyramid vertex figures and their facetings.
		- this includes those with triangular frustum vertex figures and their facetings.
	hometown.aol.com /hedrondude/polychora.html (670 words)

	The Golay Code on the Dodecadodecahedron
		Surrounding this vertex are 4 elements of this octad.
		The other 4 elements are those faces one arrives at by taking an isochromatic path from one of the 4 initial faces through the chosen vertex.
		Thus, as one traverses an edge and reaches a vertex, one continues the path by traversing across the perpendicular edge.
	homepages.wmich.edu /~drichter/golay.htm (1979 words)

	[No title] (Site not responding. Last check: )
		The right mouse button toggles the plotting between drawing all of the intermediate figures and leaving them on the screen and showing only the "current" intermediate figure.
		The bar along the top goes from its left point (where the original figure is plotted) and the right point, plotting the target figure.
		Input of the figures is from code in "myInit" (setupVertices) or from a file of points (setupVertices1) or from a calculation (setupVertices).
	www.mtholyoke.edu /~bweaver/courses/056/cs331/progs/021306_0a.cpp (196 words)

	lowyStudy
		Figures 1 and 2 are a reminder of the energy spectra and y distributions for data and MC before and after a QE-like selection procedure has been applied.
		Figures 5 show the hit multiplicities for hits that I classify as part of the vertex hadronic shower, this includes hits that are:
		Figures 8 show the numbers of shower in events and figures 9 the total number of sub-showers (of all classifications):
	www.hep.ucl.ac.uk /~med/lowyStudy.html (873 words)

	Computer Graphics - JAVA Applets by Meiko Rachimow
		A vertex figure is a polygon formed by connecting the midpoints of all adjacent sides of a vertex.
		The p describes the polygons, and the q the count of that polygons at each vertex.
		The pentatope, also called pentachoron or 4-simplex is the simplest regular figure in four dimensions.
	pentatope.de (214 words)

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