Notes on Verheyen's Symmetry Orbits

About Book Index Mistakes and suggested improvements Example of space-(n) body / system Example of conjunctive and disjunctive compounds

The first chapter of the book enumerates the symmetry groups in 3D Euclidean space. The symmetry groups are based directly on generic group algebra, not on the Schönflies groups, which are based on cristallography. This chapter is of high quality with very clear images and pictures.

The second chapter "Symmetry Action" contains a lot of abstract group algebra and is hard to understand even if you have some basic understanding of group algebra already. The problem that I had with this chapter was that it consists only of formal definitions and there aren't any examples.

In my experience all too often the academic world uses formal definitions to introduce a new concept. Although the formal definitions are important, in my humble opinion examples are the best way to introduce a new subject. This is because our brains are good at recognising patterns: we see patterns even when there is only noise. This is actually also how we built our scientific models: they are based on experiments.

Once someone has some basic understanding, the formal defintions are essential to make sure everybody will have the same understanding, I feel the need to stress this here, since I have come across this problem a lot: where the formal defintions are either given before the examples, or they aren't any exmaples at all.

I think in [HVER01], chapter 2 this classical mistake is made as well. Formal definitions of terms are given without any examples, and even without stating what this is going to be used for. With this page I hope I can provide some of examples, to help to understand what Verheyen tried to do.

It would have been helpful if the book had an index, for this reason I added one here. Note that the index below is by no means complete. It is just some of the terms that that I think can be useful to look up.

Term	Page(s)
C2 vs. D2	102
conjugate subgroups	68
conjunct bodies	72
conjunctive subcompounds	166-167
conjunctive suborbit	76
constituents	70
coorbit	75
coset	67
cyclic groups	21-22, 40-49
descriptive	70
dihedral groups	22-27, 50-57
direct isometry	16-18
disjunct bodies	72
disjunctive subcompounds	161-162
disjunctive suborbit	76
enatiomorphous coorbit	78
equipotent	67
extended groups	19-20
fundamental descriptive	84
group action (activating a set)	69-71
higher orientative stability	101
homogeneous space	69
improper subgroup	68
level-(m) body in the system	72, 74
n-body	71
maximal orientative stability	101
normal subgroup	67
opposite isometry	16-18
orbit	70
orientation	95, 100-101
orientative stability	101
oriented	101
position	74
prime	77
proper subgroup	68
realization	70
space-(n) body	72
space-(n) system	72, 83
stability	101-102
stabilizer	69, 74
subcompounds	161-167
subgroup	18, 20, 68
suborbit	75-76
superior orientative stability	101
trivial subgroup	68
μ	107
versatility	101

The listing given here are suggestions of improvements in the book, including some mistakes that I found. These are not confirmed by the author and therefore they cannot be considered as any official errata. Since the author seemed to have passed away according to this article I decided to publish them here. I hope that it will help a reader to understand the book better.

• Page 39, Figure 38: Figure titles are wrong. (a) perpendicular to a 3-fold axis (10×); (b) perpendicular to a 5-fold axis (6×); (c) occurs twice for each 2-fold axis in the image (30×); (d) perpendicular to a 2-fold axis (30×);
• Page 39, Table 16: A better title would be 'Icosahedral Angles between m- and n-Fold Axis' and then use 'n' and 'n' on top of the two left-most columns. It would also be good to refer to Firgure 38 for each angle. Between 2-fold and 5-fold axis the 90° angle is missing. It can both be seen in Figure 38b, where the 5-fold axis is perpedicular to the image and Figure 38d, where the 2-fold axis is perpendicular to the image.
• Page 52: D₄D₂ has 2 reflections, not 3.
• Page 53, Figure 56: The text 'Number of mirrors: 3 (Fig. 56a) + 1 (Fig. 56b)' could be misinterpreted. Fig. 56a itself doesn't show 3 mirrors, but there are 3 mirror planes of the type shown in Fig. 56a, where the plane contains one 3-fold and one 2-fold axis. The same improvement can be made when referring to similar images in the next symmetries.
• Page 67, Section 1.1: In the first sentence 'C' shouldn't be in bold font: it should 'C' instead.
• Page 68, item 3: It would be clearer to state that the two cosets are H and G - H
• Page 68, section 1.2: The statement A · H · A^-1 = H should be A · H · A^-1 = H ('A' shouldn't be in bold font).
• Page 69, section 1.4: In the expression A * (B · H) = (A · B) · H it is better to write A * (C · H) = (A · C) · H so the character 'C' relates to the definition of q and q'.
• Page 72: The sentence 'The sum of these components is again a space-(n-1) body and is composed of space-(n-2) bodies.' is unclear, since the 'sum of components' is undefined. If you put these components, which are space-(n-1) bodies, back in a set, then you would get back the original space-(n) body, not a space-(n-1) body. I think the author might have meant meant the union of these components, with 'the sum of'. In that case you do get a space-(n-1) body. I don't think that the Minkowski sum is meant here.
• Page 73-74: The choice of the letter I is a bit unfortunate, since it is also used to denote the central inversion.
• Page 74, Section 3, first paragraph: The expression p ∈ V(n) should be p ∈ V⁽ⁿ⁾.
• Page 74, Section 4: The first sentence would be clearer it the word 'then' is added after the first comma.
• Page 75, mapping t: G / C_s should be G / G_s. It could also be helpful to mention in the line below below for C · G_s that C ∈ G
• Page 76, first paragraph: G₁ should be G_s and p₁ should be p_s.
• Page 76, case 3: The notation ⟨H, G_s ⟩ ⊂H means that this set is used as a generator; i.e. the isometries are recursively combined with each other until no new isometries are found. This is to close the group. However this would never lead to case 1 as the text says, but it is meant that this new group becomes an F_s, which takes the role of G_s in case 1 since H ⊂ F_s.
• Page 76, bottom of the page : The symbol ⇋ is used. It would be good to mention what this symbol stands for. I think this symbol is meant to represent the 'equipotent' relation.
• Page 77: In the figure title H₁, H₂, G₁, and G₂ should be H, H_s, G and G_s respectively, I think. In that case it fits with the text as well.
• Page 78: At the top of the page there is a reference to 'Diagram 2'. That should be 'the diagram below'.
• Page 82, item 2b.: The order of C₅ indicated by h = 5 not 12.
• Page 83, Section 2.: 'distinct form' should be 'distinct from'.
• Page 125, Section 4.2: At item 1: Fig. 83 should be Fig. 81, and at item 2: Fig. 81 should be Fig. 82.
• Page 126: At the top of the page Fig. 81 should be Fig. 83.
• Page 135, Figure 100 and 101: The images show compounds of 6 cubes, not 2, i.e. 2 | A₄ x I / C₂ x I should be 6 | A₄ x I / C₂ x I (twice).
• Page 135, Figure 101: The left image of is overexposed. Figure 114 shows a better image of the same compound.
• Page 140, item 7: 2 | D₂ x I / C₂ x I should be 2 | D₂ x I / D₁ x I
• Page 157: 2n | D_4n x I / C_4n x I should be 2n | D_4n x I / C₄ x I and 2n | D_3n x I / C_3n x I should be 2n | D_3n x I / C₃ x I.
• Page 167, section 2.2.1: 2 | D₂ x I / C₂ x I should be 2 | D₂ x I / D₁ x I (all three times).
• Page 167, section 2.2.1, item a: The reference to Fig. 114 should be moved to section 2.2.2 item a.
• Page 167, section 2.2.2: 2 | A₄ x I / C₂ x I should be 6 | A₄ x I / C₂ x I (all three times).
• Page 168, Figure 114: The angle of the compound looks more like 21°30'. An angle of 14°21'33" looks more like the compound in Figure 100.
• Page 172, section 3.1, first paragraph: D₆ x I / D₃ x I should be D₃ x I / C₃ x I
• Page 172, section 3.1: Fig. 119 that is referenced to doesn't show the intended compound. It can be seen in Fig. 96 on page 128 or in Fig. 121 on the left side.
• Page 173, Figure 119: The image doesn't show 2 | D₃ x I / C₃ x I | 22°14'19".52, but the rigid 3 | D₆ x I / D₂ x I (neither does it show a compound along a 3-fold axis).
• Page 178, Figure 125: It looks like there are parts missing in the compound that is shown, though this might be because the photo is overexposed.
• Page 182: For 12A | S₄ x I / C₂ x I | 54°44'08".20 the right part of the equation 4 x 3 | D₁₂ x I / D₄ x I should be 3 x 4 | D₁₂ x I / D₃ x I. This makes the case of 6 x 2 | D₆ x I / D₃ x I trivial but such a trivial case was mentioned on page 181 in the conclusion as well.
• Page 183, last line before section 3.3: One could mention that 6 x 2 | D₈ x I / D₄ x I is shown in Figure 107, page 143.
• Page 188: 20 | A₅ x I / C₃ x I | 31°02'41".91 should be 20 | A₅ x I / C₃ x I | 23°25'51".28
• Page 192, last line before subsection 3: Also add that this is a 5 x 6 | S₄ x I / D₂ x I.
• Page 197, Figure 135: It would have made more sense to switch left and right on the bottom row. Then Bakos' compound is always on the left and the Theosophical compound is on the right.
• Page 214, Section 5.5: 4 | A₄ x I / C₂ x I should be 6 | A₄ x I / C₂ x I
• Page 224: The year 1990 referring to Brückner should be 1900

This subject is related to page 71-72. The scetion is quite abstract and some of the definitions are ambiguous. My interpretation is that Verheyen tried to use this in case a compound of cubes consists of subcompounds. A subcompound in its turn might consist of subcompounds, until you final end up with one cube. I also think that he needed to cover the conjunctive subcompounds with this.

The part that is ambiguous is when a sum of components is used. These components are elements in a set and are themselves sets as well. It is unclear what he meant with the sum. I don't think that the Minkowski sum is meant here, neither did he mean to gather the components is a the set again, I think. It might be that he meant taking the union of these components,

Let's have a look at some example, though. Imagine a cube consisting of eight vertices v₁ to v₈. You could describe the cube to consist of 6 faces, possibly as a result of an orbit of one square. Each square consists of 4 edges, which also might be the result of an orbit of one edge. The cube can then be described by the following space-(3) body:

}

This space-(3) body consists of six space-(2) bodies, which represent the faces. Assuming that Verheyen meant a union with 'the sum of these components' then this sum would lead to the following space-(2) body:

representing the twelve edges.

At level one this sum would then contain all the vertices:

Now according to Verheyen the set of these two bodies increased with the space-(3) body itself is called the space-(3) system. Here that would mean that the space-(3) system consists of the set of faces, the set of edges, and the set of vertices. The set of edges is an example of the level-(2) body in the system.

On page 76 [HVER01] makes a distinction between disjunctive and conjunctive suborbits. This leads to the disjunctive subcompounds from Chapter 5, section 1.1.1 and the discussion of conjunctive subcompounds section 1.1.2 in the same chapter.

Press the button to see an example of a compound of eight tetrahedra with disjunctive subcompounds of two tetrahedra, where the tetrahedra of the subcompounds share one vertex. Compare this with a compound of five tetrahedra with conjunctive subcompounds of two tetrahedra, where two tetrahedra share a vertex as well. In the former it is possible to make distinct pairs of these subcompounds, and hence they can be giving their own colour, while in the latter it is impossible to make these distinct pairs. That is quite obvious, since the latter is a compound of five tetrahedra. What is happening is that each tetrahedron can form a subcompound with two other tetrahedra, leading to five pairs with overlapping tetrahedra.

When you are looking for special angles for compounds with rotational freedon sometimes it is possible to find these distinct subcompounds and give them their own colour, and sometimes it isn't possbile to find distinct subcompounds. In that case constituents in one subcompound also occur in other subcompounds. Note that it is understood that in all of these cases of the subcompounds are supposed to have the same symmetry.

Since constituents appear in two or more subcompounds it is sometimes hard to see whether it is actually possible to find some combination of constituents that form distinct subcompounds. [HVER01] shows how this can be recognised by using group algebra.

The disjunctive example above is a special angle for the compound 8 | S₄A₄ / C₃ with rotational freedom. The subcompound is a 2 | D₆C₆ / D₃C₃. [HVER01] states that a subcompound is disjunctive if the subgroup, here C₃, that the descriptive shares with the final symmetry, here S₄A₄, is also contained by the symmetry group of the subcompound. In the disjunctive example given here that is indeed the case: consider e.g. the blue subcompound: the 3-fold axis that each tetrahedron in the subcompound shares with the final compound is indeed also a 3-fold axis in the subcompound. As a matter of fact it is a 6-fold axis in the subcompound.

The conjunctive example above is a special angle for the compound 5 | D₅C₅ / C₂C₁ with rotational freedom. As in the disjunctive example the conjunctive subcompound is a 2 | D₃C₃ / C₃. The subgroup C₂C₁ consists of the identity 'E' and a reflection, while 2 | D₃C₃ / C₃ consists of the rotations around a 3-fold axis and three reflections, with the reflection planes intersecting the 3-fold axis. One can see that those reflection planes are not the reflection planes that a tetrahedron shares with the final symmetry. In fact they aren't relection planes of any tetrahedron at all, since the subgroup C₃ of that subcompound only contains direct isometries. This is what makes this a conjunctive subcompound.

[HVER01] also shows how to combine conjunctive subcompounds to obtain disjunctive subcompounds. In the conjunctive example above that will lead to the whole compound, which isn't very surprising since the compound consists of five tetrahedra, and the number five is a prime number.

Consider instead this conjunctive example. This is a special angle for 12 | S₄ / C₂ for which subcompounds 2 | C₂ / E get a higher stability 2 | D₃ / C₃ because the two tetrahedra in a subcompound share a 3-fold axis. The 3D model shows four of the subcompounds in colour and it isn't possible to find more of these pairs with the remaining four gray tetrahedra. As mentioned this is a typical property of a conjunctive subcompound. Note that the dark gray tetrahedron can be combined with any green one to form another pair.

The symmetry of the conjunctive subcompounds, D₃, contains the rotations around a 3-fold axis and three half-turns, which axes are perpendicular to the 3-fold axis. For the green subcompound you can find the 3-fold axis by pulling the green vertex located just before 12 o'clock to the centre of the image. A 2-fold axis can be found between the two green vertices at four o'clock (after reloading the 3D image e.g. by pressing the button again). This 2-fold axis isn't shared with a 2-fold axis of a tetrahedron, which means that the half-turn of the subgroup C₂ from in 12 | S₄ / C₂ isn't a subgroup of the symmetry of the subcompound: D₃. Note that D₃ does have a subgroup C₂, but in this example the orientation is wrong.

[HVER01] describes that you can find disjunctive subcompounds that consist of the conjunctive subcompounds by adding the subgroup of the whole compound and to the group of the subcompound and by closing the resulting set. In the example above that means that the half-turn from the subgroup C₂ in 12 | S₄ / C₂ should be added. One of of those can be found by reloading the 3D image and pulling the part at 9 o'clock between the blue and yellow vertices to the centre of the image. Then you will look straight into a 2-fold axis, i.e. an edge, of a green tetrahedron.

If you apply this half-turn to the whole green subcompound, then the other green tetrahedron will be mapped onto the darker gray one. In other words this one needs to be added to the subcompound. Now one needs to make sure to do the same for the shared 2-fold axis of the other green tetrahedron and also for each added tetrahedron. With shared 2-fold axis is meant the 2-fold axis of the tetrahedron that is shared with the whole compound.

In this example it is enough to only add that darker gray one to close the group. If you apply this same mehod to the other colours, you will get these disjunctive subcompounds. They have the symmetry 3 | D₃ / C₂. One of the 2-fold axis can be found at the same location as indicated before when the half-turn was applied and the dark gray tetrahedron was added. The 3-fold axis is shared with a 3-fold axis of the whole compound and for the green one you can find this by taking the point between 1 and 2 o'clock and pull it towards the centre, though not all the way.

What is special for this subcompound compared to the other angles in the domain of rotatinal freedom, is that any pair of tetrahedra in the subcompound share a 3-fold axis.