Various crystallographic space groups
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4iSc2-2.png)
Only a few of these illustrations made it into The Symmetries of Things, but there aren’t such constraints here, and so I present a whole lot of pictures of the non-isotopic, or “composite’ crystallographic symmetries. Eventually I’ll add all that I have to this page, and label the figures correctly.
These were all drawn as straight up postscript programming in a few kilobytes of code that can be computed directly, as a program (!), within a laser printer.
For the best viewing pleasure, double click on an image and scroll through the lightbox.
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4iS2i-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4iSc2-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4iSk2-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4oS2o-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4oSc2-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4oSc2K-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4oSk2-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4oSk2K-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4Sb2i-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4Sbc2-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4Sbi2o-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4Sbk2-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4Sbo2o-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4tS2o-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4tSc2-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4tSc2K-2.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4tSk2-5.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/morning_glories_4tSk2K-2.png)
Match each illustration to its coupling, with the generators below. From the coupling the rest of the data follows.
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/Screen-Shot-2023-06-07-at-6.25.37-PM.png)
These symmetries fiber over the Klein bottle (!)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/lily_xbxo-2.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/lily_xixiK-2.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/lily_xoxiK-2.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/lily_xoxoK-2.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/lily_xbxb-2.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/lily_xbxi-2.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/lily_xxi-2.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/lily_xxo-2.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/Screen-Shot-2023-06-07-at-8.07.51-PM.png)
Match each illustration to its coupling, at left, and thence to its Intern. no. and Thurston-Conway fibrifold name.
632’s:
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_hexacosm.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_6t3t2o.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_6o3o2oK.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_6o3o2o.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_6i3t2d.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_6i3i2i.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_6d3o2iK.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_6d3o2i.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_63o2.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_63i2.jpg)
Exercise: There are more illustrations than couplings. Which are redundant?
Let me know.
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/Screen-Shot-2023-06-07-at-7.16.58-PM.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_Sk6k3k2-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_Sk6c3c2-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_Sk63o2-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_Sk63i2-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_Sc6k3k2-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_Sc6c3c2K-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_Sc6c3c2-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_Sc63o2-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_Sc63i2-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_S6t3t2o-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_S6o3o2o-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_S6k3k2-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_S6i3i2i-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_S6d3o2i-1.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/simple_S6c3c2-1.png)
Quiz: which three of these symmetries are missing figures?
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/Screen-Shot-2023-06-07-at-7.07.07-PM.png)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_2o2oXoK.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_2o2oXo-2.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_2o2oXiK.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_2o2oXi.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_2o2oXb.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_2o2iX.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_2i2iXK.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_2i2iXb.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_2i2iX.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/orchid_22X.jpg)
![](https://chaimgoodmanstrauss.com/wp-content/uploads/2023/06/Screen-Shot-2023-06-07-at-7.11.30-PM.png)
Match the figures with the couplings!