  == Complex Euclidean ==  Regular complex polytopes are regular, because their flags are transitive on their symmetry.  Because these exist in complex space, which has a real reflex in twice the dimension, understanding complex polytopes is a lead to understanding real symmetries in higher dimensions.  The complex polytopes are complex in the usual sense, that is, at a boundary between faces, more than faces might meet.  In normal polytopes, this is not permitted, because faces form fragments of the surface, and the surface does not divide into several sheets.  In complex euclidean space, that is, euclidean space with complex numbers, this is allowed.  A line in euclidean 2-space is given by Y = aX+b.  All of these are allowed to be complex numbers.  In Euclidean space, two planes define a point, if a and a' are different, that is, Y=aX+b, and Y=cX+d, will define a unique point, X,Y which solves both, where a <> c.    One might use a line as a mirror in complex space.  Every point, for example, lies on one of a parallel set of lines, Y=aX+c.  There is a line perpendicular to all of these, Y=-X/a or -aY=X.    A mirror corresponds to a 'rotation' in the plane Y=-X/a, of the order of w, where wn=1.  The mirror-image is of the order n, over the plane X=-aY, is effected by moving P to P', where P falls at aX+b, then P' falls at aX.   A mirror can thus create 2, 3, 4, ... images.  A kaleidoscope consists of several mirrors, with a region (tether) between the mirrors, that is reflected throughout the plane.  In complex space, a tether can join any number of mirrors, however, three mirrors is enough to create all of the groups.  A flag is a kind of tether, where the ends represent the vertex, edge-centre, etc of a polygon, polyhedron, etc.  This can be replicated over the space.  A regular polygon has a mirror through its vertex, and another through the centre of its face.  Since only two mirrors are used in a polygon, only those that need two or less mirrors are used.    A polygon like 3{4}5, has a flag or edge-vertex, with an order three mirror at one end, and an order-five mirror at the other end.  The '4' represents an alternating walk (or angle between mirrors) that is equal in both directions.  In group notation,       AAA=1       represents the order-three mirror     BBBBB=1     represents the order-five mirror     ABAB=BABA   represents the alternating walk.  Any valid string of A's and B's represents a valid walk, but these can be reduced only by the equalities above.  So, a walk of the form ABABAA = (BABA)AA = BABAAA = BAB(AAA) = BAB.  There are 1800 possible walks of this group.    All of the mirrors, except the vertex mirror, is perpendicular to the flag.  This means that the flag is in the same flat space as its image in all mirrors except for the vertex-mirror.  In complex space, this means that one can have five separate images to the plane Y=aX+b, which fall as a single line.  THE REAL REPRESENTATION  The real representation of complex space of N dimensions, is a real space of 2N dimensions.  The complex CE2 gives rise to a real E4.  A line in Complex space comes across as the Argand diagram.  One can think of such a diagram, as not just a 2d space, but one with a definite arrow around it, connecting 1, i, -1, -i in that order.  The propositions of straight lines, tells us that in 4d, when all 2d spaces are 'clifford-parallel' to each other, then two points define a 2-space, and 2 2-spaces must cross in a point (and no other way, like a line). Moreover, there is a definite angle between them.  The complex rotation around a point, which transforms X,Y to wX, wY forms a clifford parallel.  The trace of X,Y as w progresses around the unit circle tells us that in every even dimension, it is possible to comb the hairy ball so there are no calm points.  The complex polygons gives rise to figures with real symmetries.  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 120 faces, representing the real polyhedron groups with symmetries of 8, 24, 48 and 120.    