64 Lines from a Quaternionic Polytope

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Note that allowing bodies such as those in Fig. L emma 1. The argument for the reverse of a and b is similar. Geometrically, this corresponds to replacing line representations of a body with line segments. At this point, Eqs. Although this model has been shown to contribute some improvement in physical fidelity over other methods [ 33 ], stopping here would correct the standard-model trap at the expense of permitting interpenetrations [ 34 , 35 ].

To illustrate this error, let us continue with a less trivial case. Consider two bodies approaching as depicted in Fig. In this configuration, our collision detection routine identifies four potential contacts. We should not simply constrain all of these contacts, for this would result in a dual standard-model trap. It is possible to make a guess as to which contacts to enforce, but this inserts the likely possibility of generating contact forces where there are none.

Of course, we could also ignore all four contacts and allow interpenetration, but this too is clearly a nonphysical approach and leads to instabilities. An example of such a consequence is depicted in Fig. The model of Eq. Equations 15 and 16 together represent our first example of a geometrically accurate contact model. We will refer to the constraint type in Eq.

These, along with unilateral constraint, form the set of three contact constraints necessary for our model.

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In order to resolve the issues discussed in Sec. In this section, we define three fundamental contact constraints in their general form and begin to describe how they can be applied. An I -constraint deals with contacts of a single feature of one body against multiple features of another body. Here, we will define the general formulation of I -constraint for a vertex of A against multiple features of B where a subset of these potential contacts may feasibly result in a contact force at the end of the current time step.

An I -constraint seeks to. Equations 22 and 23 allow at most one contact force to be nonzero, and only permit this force to be nonzero when all other gap distances are nonpositive. Solving such a set of constraints is possible as a linear program with complementarity constraints LPCCs [ 36 ], or by introducing an auxiliary variable to convert inequalities 21 into a linear complementarity conditions before using an LCP solver [ 32 ]. This greatly reduces the number of constraints and removes the inequality of Eq. Conveniently, X -constraints can be written in the form of Eqs.

Rotation of the E8 root system

Although a X -constraint necessarily deals with contacts between the same pair of bodies, the contacts need not share features. That is, the geometric interdependencies of identified potential contacts, regarding which are to be enforced as defined by an X -constraint, may be between two otherwise seemingly independent contacts, e. Subsequently, it is conceivable for two potential contacts, neither of which have a potential force associated with them, i.

In such cases, the dynamics problem requires sufficient U - or I -constraints so as to allow generation of forces which may contribute to achieving configurations that will satisfy all constraints. This issue, however, while an important one when employing a heuristic approach to choosing primary contacts in I -constraints, is beyond the scope of this paper. In this section, we briefly discuss the nature of interpolytope contact and define a set of geometric tests that are useful for identifying and making heuristic choices regarding potential contacts. We define a convex polyhedron in terms of features Fig.

We may refer to a vertex either as an object v or as the vector v representing the vertex's position. A body is composed of a set of vertices V , edges E , and faces F defined by three vertices in counterclockwise order. These vectors are defined as. As we develop these tests, we keep in mind that a contact can occur anywhere on a polytope, but the possible normal vectors are restricted by the contacted feature.

Figures 8 — 12 depict the possible positions and orientations of contact normal vectors for each feature in two and three dimensions. Since a vertex does not have a uniquely defined normal, we use the normal from the edges that connected to the vertex. It is important to realize that when using a time-stepping method, we hope to identify contacts which will reflect these feasible contact normals at the end of the time step.

Yet, many methods of contact identification use approaches that assume continuous or static body positions. This is why we geometrically relax the following set of tests that will prove useful for identifying and making heuristic choices regarding potential contacts. Consider the simple example in Fig. If only one were included, e.

We observe that the second contact normal n 2 is outside of the normal cone region of v a 2 and does not correspond to a physically reasonable contact normal. However, this is because n 2 is determined at the beginning of the time step; yet, we wish to prevent interpenetration for the end of the time step. In order to avoid interpenetrations, we must detect and consider such contacts as C 2 , even when they are not physically feasible at the current time step. This necessity requires that we geometrically relax the normal cone region or, equivalently, relax the definition of applicability.

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The time required to solve a time-stepping subproblem, and therefore the time required to compute a simulation, is strongly dependent on the number of contacts n c. While it may seem that the number of contacts is determined by the simulation, taking a close look at an individual discrete time step, as we do in Sec. We slightly modify classical applicability conditions to derive a method to choose a small set of extra contacts.

Later, when considering potential contacts to include in a time-stepping subproblem, we will introduce a relaxed form of inequality 28 to allow for possible rotation in a time step. Whereas classical applicability is a Boolean function, our function returns a value as we may wish to use the value of applicability as a heuristic when comparing contacts. Figure 14 depicts two edges, e a and e b , in contact. Just as there are vertex—face configurations that have proximity but could not result in a force, so too are there edge—edge configurations that should not be associated with potential contact forces.

However, we can still determine edge—edge applicability in the following manner. Conceptually similar to Eq. Feasibility is similar to applicability, but is dependent on feature positions as opposed to orientations. The region of feasibility is essentially the region in which we anticipate a contact between a vertex v and a facet f could occur. We define vertex—edge and vertex—face feasibility similarly.

An example of this region is depicted in Fig. From the set of fundamental constraints presented in Sec. This abstraction layering is depicted in Fig. At the lowest level is the complementarity problem which is utilized first as a model of unilateral contact with force, but further as a tool for generating logical NAND conditions on subsets of potential contacts.

Sets of these low-level constraints can be entirely represented by abstracting to the three fundamental constraints of unilateral, intercontact, and cross-contact. In turn, the fundamental constraints are formulaically generated by the highest abstraction layer which looks only at what pairs of features between two bodies are near each other. We consider two contact configurations in 2D: vertex—edge and vertex—vertex. We do not consider edge—edge because this case is redundantly covered by the other two configurations [ 38 ].

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To be clear, edge—edge penetration in 2D is impossible if vertex—edge and vertex—vertex constraints are properly enforced, simply because edges defined and composed of vertices, and for two edges to come near each other implies that the vertices of at least one of those edges are nearing the other edge. We can formulaically generate the set of constraints for the two 2D configurations as given below.

There are four contact configurations in 3D: vertex—face, edge—edge, vertex—edge, and vertex—vertex. Again, we exclude certain feature pairs just as we did in 2D, namely, edge—face and face—face, because they become redundant when the other configurations are properly enforced. Edge—face is redundant since edges are composed of vertices, and for an edge to approach a face requires that either the vertices of that edge approach that face, or the edge approaches the edges at the perimeter of that face possibly adjacent edges , reducing to either vertex—face or edge—edge, respectively, and edge—vertex in the case that the edges of the second body are adjacent.

By a similar argument, face—face becomes redundant; a face, being composed of edges in turn composed of vertices, nearing another face implies that edges and vertices are approaching the other face. In all possible configurations, this results in one of the four feature combinations considered. The vertex—face case is analogous to the 2D vertex—edge case. Given a vertex v a near a face f b as depicted in Fig. Consider Fig. Each C ei is an edge with positive edge orientation and each C ej is an edge with negative edge orientation, and both sets of edges have applicability and feasibility with e b.

In the case of a vertex v a near another body's vertex v b as depicted in Fig. This first requires the set of I -constraints. These X -constraints are enforced only for edges e ar and e bs , which have edge—edge applicability and feasibility. The constraint in Eq. Equations 48 and 49 constitute one possible time-stepping subproblem that incorporates PEG for accurate geometric representations of angular features of interacting rigid polytopes.

As mentioned above, friction may be added easily following the discussion in Refs. The experiments discussed below compare simulations performed with the time-stepping methods defined by Eqs. The only difference in the time steppers was that one used the standard contact model and the other used PEG. For simulations with PEG, a single primary potential contact was chosen for each set of potential contacts, which yielded the least accurate form of PEG.

Despite this, the results show significant accuracy improvements when using PEG. The examples are of convex polytopes poured into nonconvex containers represented as unions of convex polytopes. Figure 20 shows a snapshot from the 2D experiment, which consisted of simulations ten different initial configurations for 16 different time steps.

One polygon was dropped every 0. At the end of each time step, the area of overlap among all polygons was computed and used as a metric of accuracy. When simulating the standard model, the simulation frequently crashed prior to reaching 5 s of simulation time, due to SM traps that caused the solver to fail. Typically prior to failure with the standard model, configurations were reached that subsequently allowed interpenetration or nonphysically large body velocities.

Pivoting solvers, such as Lemke's algorithm [ 24 ], demonstrated similar difficulties in finding physically sensible solutions in SM trap situations. By contrast, time-stepping with PEG never encountered such problems since interpenetration and overconstraining were avoided. Figure 21 shows the median and the first and third quartiles of area of overlap error for increasing time-step size.

Even for small time steps, the time stepper using PEG produced errors several orders of magnitudes smaller than the time stepper using the standard contact model. As the error with the standard model grows with the step size, the error in PEG remains imperceptibly close to machine precision. An interesting statistic to observe is the rate of occurrence of SM traps. In other words, how frequently was PEG necessary to achieve a geometrically accurate interaction between bodies. Figure 22 depicts the number of SM traps encountered at each time step for the 2D experiment.

The top curve represents the total number of contacts, and the curve below it shows the number of SM traps. One should realize that as the size of the time step increases, so would the percentage of SM traps. As shown in Fig. In the 3D simulations, the random polygons were replaced by convex polyhedra, the box became the union of five right hexahedra, and volumes of overlap replaced the error metric used in the 2D simulations. Again, there were ten sets of different initial conditions for each of 16 time-step sizes, and these sets were each used for both PEG and the standard model.

A polyhedron was dropped into the box every 0. The constraint violation error defined as volume overlap is depicted in Fig. As the step size increases, the penetrations between polyhedra grow steadily larger with the standard contact model, whereas PEG virtually eliminates this error at all time steps.

Complex polytope

The occurrence of SM traps is shown in Fig. Although the shape similarity is intriguing, it is simply coincidence, as these results are only representative of a single experiment and time-step size for both 2D and 3D. There have been previous good results regarding physical fidelity and stability of differential variational inequality DVI methods [ 10 , 12 , 42 ] upon which we have built our contact model. PEG further improves stability by avoiding interpenetrations, which can generate nonphysical behavior in ways that are difficult to predict and correct. PEG is also extendable to nonconvex bodies simply by subdividing bodies into convex parts [ 23 ].

Although PEG results in significantly smaller error while maintaining dynamic stability, there is still some increase in error for any time-stepping method as the time step increases. This is due to the fact that the time-stepping approach solves for impulses at the end of the time step based on normal directions determined at the beginning of the time step. As such, body rotations may generate unpredicted interpenetrations at the end of the time step, which are exacerbated given larger time steps.

In future work, we plan to explore efficiency and accuracy tradeoffs associated with the design of heuristics used to define the set of active contacts, the size of time steps, and the number of primary potential contacts. When simulation speed is the main goal, it is desirable to minimize the size of the active set and the number of primary potential contacts.

However, when high accuracy trumps speed, then multiple primary potential contacts per contact should be used and the active set would be larger. The pursuit of this work will require a concomitant effort in the development of efficient, large-scale solvers of linear programs with complementarity constraints. The standard-model trap. If unilateral constraints are enforced against both edges, the vertex becomes trapped by the sum of the edges' half-spaces light gray regions plus body. The same trap can occur in 3D between a vertex and set of faces or an edge against multiple edges.

Infeasibility caused by the standard contact model. Vertex v a 1 is penetrating the line containing edge e b 3 i. Similarly, v a 2 is penetrating the line containing edge e b 1 i. Since the bodies are rigid, it is impossible to eliminate both penetrations at the end of the next time step. Therefore, the time-stepping subproblem with constraint stabilization will not have a solution.

The possible values for c in the space surrounding edges e 1 and e 2. A dual case of a vertex near two edges in 2D. The physically feasible trajectories of v a are interdependent with the trajectories of v b and vice versa. A valid solution to the dual vertex—edge case using the LNC model. Note that neither vertex is penetrating the other body, perfectly satisfying the constraints of Eq. Two-dimensional normal region of an edge. Two-dimensional normal wedge at a vertex. Three-dimensional normal region for a face. The simplest of the three 3D normal regions, it is defined by a single half-space.

The Voronoi region depicted is a subspace of that half-space. Three-dimensional normal region at an edge. Again, the Voronoi region pictured is a subset of the normal region. Three-dimensional normal region at a vertex joining three faces. For the general case of m joined faces, the polyhedral region will be defined by m half-spaces planar with the faces and is equivalent to the Voronoi region for the vertex.

Vertex v a 1 has classical applicability with e b and the normal n 1 of C v a 1 , e b is within the normal cone. Vertex v a 2 does not have classical applicability with e b and the normal n 2 of C v a 2 , e b is outside its corresponding normal cone. The result in this configuration is dependent on machine precision error. Relaxation eliminates this dependency by increasing the domain of applicability. The region of feasibility for a vertex against a facet f of a body B is the region above the thick dashed line.

Talk:Complex polytope

C v a , f b is clearly the only contact to consider given a reasonable time-step size. An example of a vertex—edge configuration. If the first is not enforced, then the second must be and vice versa. A vertex v a of body A approaching a vertex v b of body B from above. A snapshot from a polygon pouring simulation. A polygon was dropped into a box every 0. At each time step, the total area of overlap between all pairs of bodies was used as the performance metric.

Comparison of area overlap errors from ten polygon pouring simulations. Predictably, time-stepping with the standard contact model generated more error for larger time steps. The error when using PEG was relatively imperceptible, set along the horizontal axis for all time steps tested.

Sample frame from a polyhedra pouring simulation. A polyhedron was dropped into a container every 0. At each time step, the total volume of overlap between all pairs of bodies was used as a performance metric. Comparison of volume overlap error for ten instances of the polyhedron pouring experiment.

Underpinning each of the complex polygons, is a group of poincare polyhedra like the poincare dodecahedron , which represent a repetition group under clifford rotations. These figures have 8, 24, 48 and faces, representing the real polyhedron groups with symmetries of 8, 24, 48 and What is the difference between the "complex line" and the complex plane? If there is no difference, why are we suddenly using the term "complex line", which I've never heard before in my life?

Claiming that boundaries don't or can't exist in complex spaces or can't even be defined strikes me as patently idiotic. Complex spaces are just like regular spaces except with half the dimensions labeled "imaginary", no?

Talk:Complex polytope - Wikipedia

It's almost as though it weren't a line at all, but rather a plane. I'd like to note that I've also never heard the term argand diagram before reading this page. I'm glad I looked it up, though, it seems quite useful, it's a way of representing the complex line as a sort of "plane". It seems strangely familiar!

How is this an edge? It's a face. Why don't you just call it a face? What is wrong with you? Also, why is it centered on the origin? Polytopes don't usually have edges or faces which include the origin, did that happen because we decided to look at an edge in isolation and we gave it its own coordinate system, and if so, how is the location of the origin of that independent coordinate system relevant in the slightest? Or is this actually R4 or something?

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I have no idea, but luckily it doesn't seem to matter because after picking a "general point" in it we wander off in mid-sentence to talk about an apparently unrelated "edge" face having p vertices. Is the "general point" one of them? It would seem not, as their locations have nothing to do with y.

Still, the math is simple enough. This is false, so oompa loompas come out and put a red bucket with a frowny face on our heads. It has 16 vertices, which for clarity have not been individually marked. Sisterly harmer talk , 10 February UTC. I'm not really into complex polytopes — particularly since they are not really polytopes at all, having no boundaries — but I'll do my best to answer your questions.

Consider this a reply from someone who originally had the same misconceptions as you do now, and is trying to show how they eventually realized what a complex space and a complex polytope really are. Usually you will see the complex line called a complex plane , as it has similarities to R 2. But you must understand that R 2 requires two real coordinates to describe a location, but C 1 requires only one complex coordinate to describe a location. While it may be a useful way of thinking about them, complex numbers are not simply ordered pairs of real numbers, but independent numbers in themselves.

Unfortunately, we are historically stuck with the term "complex plane", which as you have noted results in a complete giving up of logic when we consider C 2 , which looks like R 4 but only has two dimensions. A complex space with n dimensions is not quite the same as a real space with 2 n dimensions, though I can understand why you might think of it that way after much exposure to the undoubtedly useful Argand diagrams! But treating the real and imaginary parts of each complex coordinate defeats the whole purpose of using complex dimensions.

In particular, what you think is a 2D plane is really a 1D complex line, but it shares the real plane's property of there being no good way to define a sense of "between" given two points. Is 1, 4 between 0, 3 and 2, 5? Is it between 0, 3 and 2, 6? The set of complex numbers with imaginary part 3i? But how can this be a boundary? After all, the set of real numbers with fractional part 0. This isn't a complete analogy, because the real numbers are an ordered set and the complex numbers are not: but I hope it gets the point across. And without boundaries like the facets of a real polytope, how then do you define what is inside or outside the complex polytope?

Clearly there is no way to, and hence you must abandon all concepts of boundaries for complex polytopes — which is why they would honestly be better called "complex configurations". The n -2 n dimension distinction is the reason why you think the edges of a complex polytope are faces, because you're thinking of each complex dimension as two real dimensions.

These four complex numbers are simply points in the complex line. If this disturbs you, look at a colour wheel graph of a complex function : now each complex number is represented by a point and its colour. And yes, we gave each complex edge its own coordinate system in which it's parallel to an axis, in which case you can simply define the axes so that the complex vertices are aligned to the roots of unity. Luckily, it is fixed now, and the Oompa-Loompas are happy again.

Double sharp talk , 17 April UTC. As for why the edges are not colour-coded, I can do no better than to quote Steelpillow's comment:. Bear in mind that the edges are infinite in extent and the coloured regions are not bounded in the actual polytope which is really an infinitely-extending configuration and not a bounded polytope at all , so that would need to be explained. The edges overlap in the diagram and so colour-coding would be difficult. Additionally they are infinite and therefore any colour-coding perhaps of the real-square regions would require another explanation.

Now do you see the difficulty in making the diagram any clearer? It would be great to be able to put people into a universe with true complex dimensions, and I would love to see such a thing for myself, but unfortunately I doubt life as we know it would be able to survive in such a universe, which is quite a disappointment, if you ask me. Although yes, I do think some more explanation on where the complex vertices are in the diagram wouldn't hurt. The description of Coxeter's notation appears to be garbled, as well as badly written.

Is anybody in a position to correct and clarify it? In a real regular polygon, each vertex is shared by two edges, and each edge connects two vertices. Hence we generalise a complex edge with n vertices to be part of a complex line passing through the n th roots of unity. Here we must clear up a potential source of misconceptions. In an Argand diagram, these vertices of the complex edge in question look like the vertices of a regular real n -gon.

However, one crucial point about a real polygon is that its sides are bounded. Thus we can have the inequalities that can be used as the definition of a real convex polytope. Hence no ordering is possible. Hence, we have to take the whole line instead of some part of it. Yes, I admit that I lied slightly in the previous paragraph to simplify things.

Your forgiveness and understanding is requested. We can now go up a dimension. Further, q is the length of the minimal cycle of vertices needed for every consecutive pair but not triple of vertices lie on an edge. For real polygons injected into C 2 , q is simply the number of sides. The definition of a regular polytope having a flag-transitive automorphism group generalises accordingly from R n to C n and H n. Any vector space over a field or skew field will work, if I understand this correctly. In all three cases, this automorphism group may be generated by reflections, and thus the regular real, complex, and quaternionic polytopes have been classified.

Double sharp talk , 13 April UTC. Firstly, it is simply not good enough to offer an unsupported definition in the lead, only to be forced to explain that there is no such definition. I have already explained all this and there are plenty more policies and guidelines along similar grounds.

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  7. It is not negotiable. Secondly, there was a confusion over the significance of convexity theory, as if it was somehow the fount of all real polytopes. There is a large body of mathematics concerned with star polytopes , polytopes with non-spherical Euler numbers , wholly abstract polytopes and so forth, and the presentation needs to be intelligible in that context.

    I did not look further into it as it was too major an edit to untangle. Coxeter-Dynkin diagrams have recently been added, without sourcing. Is there a source for their use with complex polytopes? If not, this is original research WP:OR and needs to be reverted. The addition has also changed the meaning of the phrase "the modern notation" to apply to these diagrams instead of that attested in the sources. Whatever the case with the diagrams, the status of Coxeter's own original notation needs to be made clear. A recent sequence of edits states that all complex polytopes are configurations. In general this is untrue, only the regular variety are configurations.

    This is because the definition of a configuration demands the same high degree of symmetry that is seen only in the regular polytopes - the same number of lines at each point, the same number of points on each line, etc. These edits are now buried behind a further sequence, so this all needs unpicking again.

    The tables have got too complicated and in anything but the widest screens they begin to squash badly. The Dynkin symbols are the first suffer. It is worst with the widest tables, such as the list in five dimensions. A better approach is needed. Either the information for each entry must be drastically cut back, or the idea of listing them all in the root article abandoned. User:Steelpillow made the image Image:ComplexOctagon. Since User:Steelpillow complained about my editing his image and reverted so I made a "2" version for this article, moving here for discussion.

    I also added the right perspective image to help clarify the global topology of the polygon. The primary issue I have with the original left-most image uncolored and no node markings is its less clear what the squares are, or how they're connected, or the ambiguous meaning of the intersecting lines. Here's a reproduced image from Coxeter, p. So the 4-edges are "unfolded" into 4 line segments in each row and column. I added blue and red colors, and used Coxeter's vertex labeling. Here's a quick retyping from pp. On your questions of interior of a polytope, that is a different issue than the interior of an edge, or k-edge.

    A star polygon has no interior, but its edges can still be drawn with a solid interior. A one-dimensional real polytope is often regarded as a closed line segment - in Plato's words it is "solid".