Beyond the fourth dimension

Coxeter, like Forsyte's character "Old Jolyon", "ached a bit from sheer love of it all, feeling perhaps, deep down, that he had not very much longer to enjoy it. The thought that some day - perhaps not ten years hence, perhaps not five - all this world would be taken away from him, before he had exhausted his powers of loving it, seemed to him in the nature of an injustice, brooding over his horizon. If anything came after this life, it wouldn't be what he wanted."
Commentary on the last years of Coxeter's life in "King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry", by Siobhan Roberts.

What happens at dimensions higher than 4?

Regular polytopes

Before moving on, we need new terms when discussion polytopes in many dimensions: The general equivalent of a face in a polyhedron and a cell in a polychoron is a facet. This is the highest-dimensional element of the surface of a n-dimensional polytope (henceforth, a n-polytope), thus having n−1 dimensions. This is also the case for the vertex figures. The ridges are n−2-D elements of the n-polytope's surface. This is also the case for the edge figures. In a regular polytope, all surface elements of a particular dimension are not only identical, but the polytope also looks identical as seen from each instance of those elements. This implies that each of the n types of element corresponds to a particular overall n−1-D symmetry of the polytope.

In Euclidean spaces with more than 4 dimensions, the only regular polytopes are the aforementioned simplices, hypercubes and their duals, the cross polytopes, or ``orthoplexes", making therefore a total of 3 regular polytopes for each Euclidean n-dimensional (henceforth n-D) space. These polytope families are infinite, with members in all n-spaces. Apart from them, there are no other regular polytopes. There are no analogues of Icosahedra/Dodecahedra or 600-cell/120-cell, there are no regular star polytopes either, and nothing like the 24-cell, which exists only in 4-D Euclidean space. In this way, one comes to appreciate the fact the latter are exceptional objects.

We now enumerate the basic characteristics of these families that will be important for understanding what follows.

Regarding the simplices, they always have the smaller number of elements possible for a polytope in their n-space. The n-simplex (e.g., a Tetrahedron, ...) is built from a n−1-simplex A (... a Triangle) with side ℓ by adding a single point outside A's n−1-D hyperplane, at distance ℓ from all vertices of A (... a pyramid apex, where Triangle A is the base), and linking that point to all the others with n other n−1-simplices (... n = 3 additional Triangular faces). Thus, the n-simplex has n+1 vertices, n+1 facets, and both the facets and vertex figures are n−1-simplices. They are therefore self-dual. They have no central symmetry.
The n-orthoplex (e.g., an Octahedron, ...) is built from a n−1-orthoplex A (... a Square) with side ℓ by adding two points, P1 and P2, in a line perpendicular to A's hyperplane at distance ℓ from A's vertices, one of them above and another one below A's hyperplane. Then those points are linked to A's vertices using n additional n−1-simplices for both P1 and P2, thus doubling the number of facets (... 2 × 4 = 8 Triangular sides). Importantly, P1 and P2 are not directly linked to each other. In this process A becomes an equatorial polytope. This implies that orthoplexes have 2n vertices and 2ⁿ facets.
The n-dimensional hypercube, or n-cube (e.g., a Cube, ...) can be built from a n−1-cube A (... a Square, with 4 sides) with side ℓ by adding an identical n−1-cube, B, at distance ℓ from A along a perpendicular direction (... a second Square); this doubles the number of vertices. Then, A and B are connected with n−1-cubes, as many as the number of facets of A and B (... four additional Squares), in hyperplanes perpendicular to those of A and B. Thus, the number of facets grows by two with each dimension, their number is 2n; the number of vertices is 2ⁿ.

The n-orthoplex is the dual of the n-cube. They therefore share the same symmetry, which includes central symmetry. The n−1-simplex facets of the former correspond to the n−1-simplex vertex figures of the latter. The n−1-hypercubic facets of the latter correspond to the n−1-orthoplex vertex figures of the former. Since only the latter elements have central symmetry, only the orthoplexes have equatorial polytopes and only the Hypercubes have the dual equatorial facet rings. The n-orthoplex has an equatorial n−1-orthoplex between any two opposite vertices, their number is therefore 2 n / 2 = n.

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We now present some Zomable projections of higher dimensional regular polytopes into three dimensions. In all these models, the white connectors indicate real vertices, the other colours denote edge intersections that happen because of the projection, not in the polytopes themselves, because they are all convex.

We start with the simplexes. When projecting a n-simplex into 2-D space, we can always arrange the direction of the projection in a way that the vertices are all equidistant from the centre. The reason is that, as we've seen above, when we build the n+1-simplex, we add a new vertex that is equidistant from all the vertices of the n-simplex. Thus, projecting the n-simplex into 2-D space along the axis that would link it to the new vertex of the n+1-simplex all points must lie at the same distance from the centre. Their distribution would be a regular Polygon with n+1 sides. To project the n+1-simplex, there are two easy options: one of them would be to project the new vertex to the centre (i.e., using the same projection vector as the previous projection), or project all vertices as the vertices of a n+1 Polygon.

For projections into three dimensions, the logic is the same, except that we can't find symmetries with an arbitrary number of vertices. For instance, for the 4-simplex, which has 5 vertices, there is no symmetric distribution of all vertices around the centre: what was done in Fig. 5.3 was to use a symmetric distribution of vertices (of the Tetrahedron) and project the new vertex to the centre. The other Zomable projections of the 5-cell (Fig. 7.8) are not symmetric.

However, the 5-simplex has 6 vertices and 6 facets; this means that we can choose the direction of the projection in such a way that the vertices are distributed as the vertices of an Octahedron, without superpositions.

Fig. 9.1: An Octahedral projection of the 5-simplex.

In all simplices, the edges connect all vertex pairs, and in this model they are also shown without superposition. Thus, the model in Fig. 9.1, which is an accurate model of the Octahedral projection of the 5-simplex to three dimensions, resembles the vertex-first projection of the 16-cell in Fig. 5.4, with the difference that there are no vertices at the centre.

The 6-simplex has 7 vertices and 7 facets. For this, we can make a projection that uses the same projection vector as the projection in Fig. 9.1, this will place the additional vertex at the centre of the model. That projection would then be identical to that of the vertex-first projection of the 16-cell in Fig. 5.4. There would be 6 new blue edges connecting that vertex to all others, which will be superposed on the three blue edges of the previous model. This would be an analogue of the projection of the 5-cell in Fig. 5.3.

By the same logic used in the 5-simplex, it is easy to make a symmetric projection for the 7-simplex, which has 8 vertices and 8 facets. We can choose the direction of the projection in such a way that the vertices are distributed as the vertices of a Cube, without superpositions.

Fig. 9.2: An Octahedral projection of the 7-simplex.

Again, the edges connect all vertex pairs. Thus, the model in Fig. 9.2, which is an accurate model of the Octahedral projection of the 7-simplex to three dimensions, resembles the model of the Cube and its faceting in Fig. 4.5b, with the difference that there are four yellow edges connecting pairs of opposite vertices.

Using the same projection vector used in the model of Fig. 9.2, the 8-simplex can also represented by adding a vertex at the centre and connecting it with all others with yellow struts. These would then be superposed with the previous yellow edges. The projection would then look exactly the same as the projection in Fig. 9.4. The same strategy can be applied for the remaining projections of n-simplexes without central vertices to make projections of n+1-simplexes.

The 11-simplex has 12 vertices and 12 facets. We can choose the direction of the projection in such a way that the vertices are distributed as the vertices of an Icosahedron, without superpositions.

Fig. 9.3: An Icosahedral projection of the 11-simplex. This model required 30 R3 struts, but this can avoided by building the model on a smaller scale, as in Fig. 4.6c.

Again, the edges connect all vertex pairs. Thus, the model in Fig. 9.3, which is an accurate model of the Icosahedral projection of the 11-simplex to three dimensions, resembles the model of the Icosahedron and its facetings in Fig. 4.6c, with the difference that there are 6 red edges connecting pairs of opposite vertices.

One could in principle make additional Icosahedral projections of the 19-simplex, for which the 20 vertices would have the distribution of the vertices of the Dodecahedron and have 10 yellow edges connecting pairs of opposite vertices, or of the 29-simplex, for which the 30 vertices would have the distribution of the vertices of the Icosidodecahedron and would have 15 blue edges connecting pairs of opposite edges. Their models would resemble, respectively, the models of the Dodecahedron and its facetings (Fig. 4.8) plus yellow radials and of the Icosidodecahedron and its facetings (Fig. 4.12) plus blue radials. However, as we'll see below, these projections are not Zomable.

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The same idea used to make symmetric projections of the simplexes can also be used to make symmetric orthographic projections of the orthoplexes. The reason is identical as for the simplexes: when we build a n+1 orthoplex from a n-orthoplex, we add two new vertices that are equidistant from all previous vertices. This means that, when projecting the n-orthoplex along the axis that would link it to the two new vertices of the n+1-orthoplex, all points must lie at the same distance from the centre. Projecting the n+1 orthoplex along the same projection vector would add the two new vertices to the centre of the projection. The vertex-first projection of the 16-cell in Fig. 5.4, where two new vertices were added at the centre of an Octahedron, is of this kind.

The other type of projection has no vertices at the centre. The cell-centered projection of the 16-cell on the lower left of Fig. 5.4 is of this kind. Its 8 vertices are arranged symmetrically, which implies they are arranged as the vertices of a Cube. As remarked then, this projection is especially good for showing all the vertices separately. This makes it possible to see that the edges - all shown without superpositions as well - connect all vertex pairs except opposing vertices, as in all orthoplexes (this can more easily be seen in the Octahedron).

Using the same logic as in the vertex-first projection of the 16-cell, we can represent the 5-orthoplex by using the same projection vector as in the cell-first projection of the 16-cell and adding two new vertices, which would appear superposed at the centre (see Fig. 9.4). These don't connect to each other, but connect to all others via yellow struts. The model would look exactly the same as the vertex-centered projection of the 8-simplex. The same strategy can be applied for the remaining projections of n-orthoplexes without central vertices to make projections of n+1-orthoplexes.

Fig. 9.4: An Octahedral projection of the 5-orthoplex.

Fig. 9.5 shows the 6-D analogue of the cell-first projection of the 16-cell in Fig. 5.4, the Icosahedral projection of the 6-orthoplex. In this projection, the 2 × 6 = 12 vertices are arranged symmetrically, which implies they are arranged as the vertices of the Icosahedron. Again, all vertices are shown without superpositions. This allows us to see that, again, the edges, which are also shown without superpositions, connect all vertex pairs except opposing vertices. This makes the model different from the model of the 11-simplex in Fig. 9.3, where the red radials were fulfilling that role, but identical to the model of the Icosahedron and its facetings in Fig. 4.6c.

Fig. 9.5: The Icosahedral projection of the 6-orthoplex.

This polytope has 60 edges (the 30 of the outer Icosahedron plus the 30 of the inner Stellated dodecahedron), 160 Triangular faces, 240 Tetrahedral cells, 192 5-cell ridges and 2⁶ = 64 5-simplex facets.

In the Icosahedral projection of the 10-orthoplex, the 20 vertices will be distributed as the vertices of a Dodecahedron, and the edges will connect all vertex pairs, except opposing vertices. Thus, the projection will look very similar to the model of the Dodecahedron and its facetings in Fig. 4.8, with B and G struts, only with the difference that instead of a Compound of five tetrahedra, a Compound of ten tetrahedra would be needed. However, as remarked after Fig. 4.7b, this compound is not Zomable.

The Icosahedral projection of the 15-orthoplex would have its 30 vertices distributed as the vertices of the Icosidodecahedron. The edges connect all vertex pairs, except opposing vertices; the resulting projection would look somewhat like the model of the Icosidodecahedron and its facetings in Fig. 4.12, except that it would require three additional edge directions with are not Zomable.

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For hypercubes, the reasoning above no longer applies: when building a n+1 cube from a n-cube, not all vertices of the n-cube are equidistant from the new vertices of the n+1 cube. For that reason, the strategy will be different in this case.

Several projections of higher-dimensional hypercubes are not only Zomable, but also Zomable in a single colour, with the models having a high degree of symmetry. The reason for this can be understood from the study of the models of the Cube and Tesseract. Since an orthographic projection is an affine projection, two parallel edges of any polytope are still parallel in their 3-D projections. This means that for a n-cube, where there are n distinct edge directions (which can be made to be the n distinct orthogonal axes defining that space), the projection will also have n distinct edge directions.

The three axes of the Cube can be represented in 3 dimensions without distortion, in three orthogonal directions, all with blue struts. If we want to project it in two dimensions, we can choose to project all edge directions equally, in which case they must have a symmetrical Triangular arrangement (see Fig. 5.2, model B). All edges are represented with the same length, and all vertices connect to three edges, which represent the full set of edge directions. Similarly, for the Tesseract, we cannot represent the four axes as orthogonal directions in 3-D space, but we can choose to represent them equally, which implies a symmetric - in this case Tetrahedral - arrangement. This is the vertex-first projection of the Tesseract in Fig. 5.4, which is the dual of the face-first projection of the 16-cell in the same figure. All edges are represented with the same length, and all vertices connect to four edges, which have the full set of edge directions.

For the 6-cube, we can also make an orthographic projection to three dimensions and treat all edges directions equally (implying also symmetrically) by making them parallel the six axes of symmetry of the Icosahedron, which are represented in the Zometool by the R struts. A model of this projection is shown in Fig. 9.6. This is the dual of the projection of the 6-orthoplex in Fig. 9.5.

Fig. 9.6: An Icosahedral projection of the 6-cube, the dual of the 6-orthoplex. This is the dual of the Icosahedral projection of the 6-orthoplex in Fig. 9.5. I made this model with the help of a vZome model made by Scott Vorthmann.

All elements are the duals of the elements of the 6-orthoplex in Fig. 9.5. The 12 5-cubic facets correspond to the the vertices of the dual model, like them their projections are all identical and are associated with the 6 axes of 5-fold symmetry of the model. The 60 Tesseractic ridges are perpendicular (in 6 dimensions) to the edges of the dual model; the 160 Cubic cells correspond to the 160 Triangular faces of the dual, the 240 Square faces, which are all projected in the same way (as Golden rhombuses), correspond to the 240 Tetrahedral cells of the dual, the 192 edges are perpendicular to the 192 5-cell ridges of the dual and the and 64 vertices correspond to the 64 5-simplex facets of the dual. The Golden rhombic faces are analogous to the faces of the vertex-centered projection of the Tesseract, which are projected as Yellow rhombuses.

Note also how much larger the projection is relative to the size of the edge, a phenomenon we had already see in Figs. 5.2 and 5.4. However, unlike those projections, this is not a vertex-first projection. In this model, we start to see that projecting objects from dimensions higher than four into three dimensional space leads to a concentration of vertices and edges near the centre.

Fig. 9.6a: The Great rhombic triacontahedron.

To build the model in Fig. 9.6, we start by building a core with the vertices and edges of the Great Rhombic triacontahedron (Fig. 9.6a). This pair of polytopes appears in the cover of Coxeter's ``Regular Polytopes".

Any orthographic projection of a hypercubic element into a plane will be circumscribed by a polygon with central symmetry, where, as in the original hypercube, each edge has an opposite edge that is parallel to it. Any orthographic projection of any n-cube will therefore be bound by a polyhedron with this type of faces: the aforementioned zonohedra. The characteristics of their faces imply that all faces are part of face rings, each containing a particular edge direction that does not exist outside that ring, each of these corresponds to one of the directions of the edges of the hypercube.

In the case of the projection in Fig. 9.6., the envelope is a zonohedron with Icosahedral symmetry that is constructible with R struts and has Golden rhombic faces, which are projections of Square faces. We have seen this object before, it is the Rhombic triacontahedron. This situation is analogous to the envelope of the vertex-first projection of the Tesseract in Fig. 5.4, a Zonohedron with Octahedral symmetry and Yellow rhombic faces (also the projections of Square faces) that can be represented with Y struts, the Rhombic dodecahedron. Just as the vertex-first projection of the Tesseract showed the parallelism between the yellow radials of the Cube (Fig. 4.3a) and yellow edges of the Rhombic dodecahedron, the model in Fig. 9.6 shows the parallelism between the red radial edge directions of the Icosahedron (Fig. 4.2) and the red edges of the Rhombic triacontahedron.

Fig. 9.6b: My Big Ball of Whacks (6-color edition), with each colour showing five of the ten faces in each equatorial ring. Each colour is associated with a single edge direction, which is perpendicular to that ring.

There is a magnetic toy puzzle that can be arranged as a Rhombic triacontahedron, the Big ball of whacks. The pieces come in six colours. At first I was wondering why six: with five colours, it would have been easier to create symmetrical patterns. However, I could find a pattern where the five Golden rhombuses of each of the six colours covers half the faces of a particular equatorial ring of 10 faces (each of the 30 faces belongs to two rings). For each of these colours/rings there is a single edge direction that is common to all the faces in the same ring, and perpendicular to the ring itself (see Fig. 9.6b); this edge direction is absent outside this ring. This is a very nice general illustration of the concept of a Zonohedron, and in this case of the 6 edge directions of the projection of the 6-cube in Fig. 9.6.

The same logic can be applied to projections of some higher-cubes. An an example, if during an orthographic projection to 3-D space we treat all 10 edge directions of the 10-cube equally and therefore symmetrically, we must make them parallel to the 10 axes of 3-fold symmetry of the Dodecahedron. Since these are represented in the Zometool by Y struts, this results in a yellow projection of the 10-cube with full icosahedral symmetry (see Scott Vorthmann's vZome model here). This projection is so complex that it is impractical to make a physical model of it with the Zometool. However, its envelope is simple (see Fig. 9.7); it is the largest Zomable yellow zonohedron, the rhombic enneacontahedron. This has 10 edge directions, and thus 10 face rings, 5 yellow rhombuses around each of the 12 vertices in the 6 axes of 5-fold symmetry, and 30 long yellow rhombuses centred on the axes of 2-fold symmetry.

Fig. 9.7: The Rhombic enneacontahedron. This zonohedron is the envelope of the Icosahedral projection of the 10-cube.

If, in an orthographic projection of the 15-cube to 3 dimensions we treat all its 15 edge directions equally, we must make them parallel to the 15 axes of 2-fold Icosahedral symmetry, resulting in a blue projection of the 15-cube with full icosahedral symmetry. Its envelope is the largest blue zonohedron that can be represented with the Zometool, the Truncated icosidodecahedron.

This logic continues for higher dimensions: we can still make symmetrical models, but now we can no longer treat all edge directions equally. Any Zomable zonohedron corresponds to a Zomable projection of a Hypercube. Given that the Zometool connector has 31 axes of symmetry (6 red, 10 yellow and 15 blue), it can theoretically represent an Icosahedral projection of a 31-cube. While that projection itself is practically impossible to make, its zonohedral envelope is relatively simple.

Finally, we should also remark that these higher-cubes can be projected as lower-D hypercubes. Thus, a simple Cube in 3 dimensions also represents a particular projection of each of the Hypercubes with dimensions higher than 3 where all other dimensions are reduced to points. The same applies to all other models of hypercubes in this page.

Semi-regular polytopes

As we remarked when discussing the partially regular polychora, associated with the orthoplexes are the non-convex demi-crosses. The facets of the n-orthoplex are half of the facets of the n-orthoplex (2ⁿ n−1-simplices) and its n equatorial n−1-orthoplexes; the vertex figure is the n−1-demi-cross; all other elements are as in the n-orthoplexes. This works because, as mentioned above, the vertex figures of the n−1-simplices and orthoplexes are, respectively, n−2-simplices and orthoplexes. The demi-crosses have a special characteristic of being non-orientable.

There is a related infinite family of convex uniform polytopes, the aforementioned demi-cubes. The demi-crosses are obtained by removing alternate facets of orthoplexes; the demi-cubes are convex polytopes obtained by a dual operation, removing alternate vertices of the hypercubes, the duals of the orthoplexes. This implies that the demi-crosses share the symmetries of the demi-cubes.

The demi-hypercubic family is thus far hidden because, as we've seen here, all uniform polyhedra and polychora that have their symmetries already exist as members of other families, two of them being regular: the demi-cube is the Tetrahedron and the demi-tesseract is the 16-cell. However, as also discussed in detail here, there is a special relation between these symmetries and the Zometool, because, among the polytopes built from the octahedral, tesseractic and icositetrachoric symmetries, only the full set of uniform polytopes that can also be built with demi-cubic and demi-tesseractic symmetries, plus the Cube and Tesseract, are Zomable. Furthermore, as also mentioned before, the demi-tesseractic symmetry generates an interesting phenomenon, triality: this means that these polychora have either 3 or 9 Zomable projections each!

It is only in 5 dimensions that we find new polytopes that have the demihypercubic symmetry as their highest symmetry. Of special interest is the semi-regular demipenteract, which has 10 16-cells and 16 5-cells as facets. But, despite the fact that this family is infinite, the list of demi-cubes that are either regular or semi-regular ends here! Interestingly, all n-demi-crosses (which have the same symmetry) are not only semi-regular, with simplices and orthoplexes as facets, but also (as defined in this site) partially regular: a faceting of a regular polytope where all instances of all elements up to the ridges are identical to those of a regular polytope, only the regular facets are of two kinds.

This similarity of the demipenteract with the demi-crosses marks it as a member of a family of semi-regular convex polytopes with the same behaviour, the k₂₁ polytope family. Like the demi-crosses, this family exists because, as mentioned above, n-simplices and n-orthoplexes have, respectively, n−1-simplices and orthoplexes as vertex figures. The 0₂₁ polytope is the Rectified 5-cell - a semi-regular polychoron with simplices (Tetrahedra) and orthoplexes (Octahedra) as cells. This is the vertex figure of a 5-D object with 4-simplices (5-cells) and 4-orthoplexes (16-cells) as facets: the 1₂₁ polytope, the demipenteract.

It is only in 6 dimensions that the k₂₁ polytopes fully emerge, with unique new symmetries. The 2₂₁ semi-regular polytope has the demipenteract as vertex figure; its facets are 72 5-simplices and 27 5-orthoplexes. It is one of 39 uniform polytopes with E6 symmetry. In seven dimensions, the 3₂₁ semi-regular polytope has the 2₂₁ polytope as vertex figure; its facets are 576 6-simplices and 126 6-orthoplexes. It is one of 127 uniform polytopes with E7 symmetry. Finally, in eight dimensions, the 4₂₁ semi-regular polytope (see Figs. 9.7 and 9.8) has the 3₂₁ polytope as its vertex figure; its facets are 17280 7-simplices and 2160 7-orthoplexes. It is is one of 255 uniform polytopes with E8 symmetry. All E8 polytopes have 696,729,600 symmetries!

Fig. 9.8: A Zomable projection of the 4₂₁ polytope.
This model required 120 B3s, 60 R3s and 120 Y3s. Alternatively, the model can be made on a smaller scale.

Fig. 9.8 shows a Zomable projection of the 4₂₁ polytope that has Icosahedral symmetry, and shows that it is a sub-symmetry of the E8. The same happens with the Hexacosichoric symmetry of the 600-cell, this is highlighted here by the fact that the 240 vertices project here as the vertices of two concentric 600-cells, with one of them being larger than the other by a factor of φ. It is not possible to project all 6720 edges into a 3-D Zometool model: in this projection, they are all Zomable, but the set of strut intersections is just too dense. Therefore, we must choose a subset of edges to represent. In this model, designed by David Richter, he used a beautiful fact, that the Icosahedral projection of the 4₂₁ polytope also contains all 2 ×720 = 1440 the edges of those two concentric 600-cells.

For more on the the Zometool models of the k₂₁ polytopes, especially their Zomable projections, check this vZome page. For an Icosahedral projection of the 4₂₁ polytope with all edge directions represented, see Fig. 9.9.

Fig. 9.9: An orthographic projection of the 4₂₁ polytope in 3-D space, now with all the edges represented. This model is laser-engraved in glass.

This was a present from one of my students, Jose Martinez.

That's it! Unlike in the case of the demi-crosses, the family ends, because its next member has an angular defect around each vertex of zero, being therefore a tesselation of the 8-D space, the 5₂₁ honeycomb. The fact that it is infinite implies that it cannot serve as a vertex figure for any higher-D polytope. From this, we conclude that the k₂₁ polytopes, and the E6, E7 and E8 symmetries are also exceptional objects.

The polytopes mentioned in this page have a complex web of geometric relations, which can be better understood using their Coxeter-Dinkin diagrams. They have many other truly wonderful properties. For the interested reader, I recommend Coxeter (1973) and (1991). There is more to explore, much much more!

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