# Clicking ”symmetri”

```Taking a walk in
the Garden of Knowledge
Speaker: Ambj&ouml;rn Naeve
Affiliation: Centre for user-oriented IT-Design (CID)
Dept. of Numerical Analysis and Computing Science
Royal Institute of Technology (KTH)
Stockholm, Sweden
What follows are some snapshots
from a walk in the third prototype of
the Garden of Knowledge.
This prototype was developed at CID during 1996/97
in collaboration with the Royal College of Music
and the Shift New Media Group.
The prototype is available on CD-rom
and part of it is accessible on the web
The entrance to the Garden of Knowledge
The overall subject patch
Clicking ”geometri” opens the geometry patch.
Overall view of the geometry patch
Browsing the geometry patch
Clicking the left margin returns to the overall subject patch.
Pointing to ”symmetri” produces a definition
Symmetry = invariance under motion.
Clicking ”symmetri” opens a new sublevel
Pointing to ”rosetter” shows preview of content.
Pointing to ”band” shows preview of content
Clicking ”band” opens a new sublevel
Clicking ”make your own bands” brings up a tool
which lets you create bands according to the 7 different symmetry types.
Clicking on ”exercises” lets you practice
and guess which symmetry elements that are present in a chosen band.
choose a band
Exercises
Is there a horizontal reflection?
Is there a vertical reflection?
Is there a half turn?
Is there a glide reflection?
Clicking ”l&auml;nkar” leads to musical symmetries
Clicking the notes plays the pitch symmetry
pitch symmetry
(without rhythm symmetry elements)
Clicking ”rytmsymmetri” adds symmetries of rythm
(with the possible choices of translation or reflection)
Clicking ”f&ouml;rdjupning” gives deeper explanation
of why there are 7 different types of band symmetries. Clicking
“band” prompts you to try to generate the two missing symmetries
by choosing different combinations of L, V, G, H.
Active
area
Click the band to activate / clean it.
Here we have found the 7:th symmetry type
In ”regler” we check all combinations of L,V,G,H
showing which combinations that give valid symmetry types.
rules
rules:
And which combinations that break the rules
rules:
In “bevis” we prove that L,V,G,H generate the symmetries
Step1: Showing that a plane isometry is determined by how it maps 3 points.
Step2: Showing that such a mapping can be achieved by ≤ 3 reflections in lines.
proof
Step 3: proving a lemma
Lemma: Reflections in two lines L1 and L2 is invariant under
rotation of the lines around their point of intersection.
Step 4: applying the lemma to reflection in 3 lines
First: Rotate the two first lines so that the second is perpendicular to the third.
Second: Rotate the two last lines so that the second is parallel to the first.
Conclusion: Reflection in three arbitrary positioned lines is a glide reflection.
Clicking “glidspegling” shows a glide reflection
Clicking “tapeter” opens a new sublevel
(tapeter = wallpapers)
Clicking “make your own wallpapers” brings up a tool
which lets you experiment with the 17 different wallpaper symmetries.
Clicking “Alhambra” takes you to a web-site
which contains examples of all the 17 symmetry types in islamic art.
One of the patterns displayed at Alhambra
Another one of the patterns at Alhambra
```