 Symmetries,
Patterns & Tessellations
Constructed With The Geometer's Sketchpad
by Allan Bergmann Jensen, original version 1997. Updated March 1999.
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Contents:
- Download
All tessellation's and guides showed in the gallery can be downloaded from here...
- Link and references
The comment on this page is in Danish, but all links will take you to
homepages in English.
Most topics presented on these WebPages can be found on WebSites written
in English. See my listing on the link and referencepage.
All links will take you to sites written in English! This WebSite is (according to my
knowledge) the first Danish WebSite covering this specific topic and the aim is to present
GSP and the symmetry for Danish teachers and Danish students.
My main point is that dynamic geometry should be presented for the student
primarily through working with basic symmetry: Reflection, rotation, transformation and
(perhaps) glidereflection. Many of the fundamental geometric figures, like squares,
rectangles and polygons, are often best constructed (and perhaps even best understood)
through the use of symmetry. When working with dynamic geometry, the dynamics of a
figure should, whenever possible, reflect the inner symmetric qualities of the particular
figure. So, dragging symmetric points, should result in symmetric behaviour.
What is the most basic feature of a square?
Most people will give an answer something like this:
Four segments of the same length, arranged with four corners each 90 degrees.
But perhaps this kind of definition is the result of teaching and culture.
We can imagine the person giving this answer constructing the square using a protractor
(or compass) to raise the perpendicular lines and a compass or a
ruler to mark the length of. the sides.
But consider an answer like:
Segments between the images of a point rotated 90 degrees 4 times in another point.
The latter explanation uses a few more words. But the thinking and
construction isn't difficult. Try out for yourself.
Symmetries, patterns and tessellation's is a superb subject for teaching
basic geometry because:
It is visual and the results are spectacular - good patters scream:
WATCH ME!
Many "rules" can be formulated and discovered by the students.
It is easy to teach more than one level at the same time and the levels
go from simple jigsaw exercises to complex group theory.
It is appealing, even to the student who is less mathematical in his/her
thinking.
The subject seems to be endless - and there are unsolved problems for
students who seek immortality (f.ex. how many pentagons can tile the plane? In 1985
fourteen types were known).
The skills in analysing pattern might be useful in many different
studies.
The student's performance in geometry is more effective when he/she
takes advantage of the symmetries in constructions.
Wallpaper groups with GSP
Download a collection of GSP-sketches illustrating all the 17-wallpaper groups:
wall_1.exe (self extracting zipfile, 263 kb,
Windows 3.1 or later, GSP v.3)
Construction help
Download a collection with construction help regarding wallpaper patterns with
GSP:
wall_2.exe (self extracting zipfile, 160 kb,
Windows 3.1 or later, GSP v.3)
Penrose tiling
Download a collection of GSP-scripts, which can be used to construct penrose
tilings (Fat rhomb and thin rhomb). Extract the files to your "Script Tool
Directory".
penrose.exe (self extracting zipfile, 90 kb,
Windows 3.1 or later, GSP v.3)
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