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Topic: T, TR, TV, TG, THG, TRVG, and TRHVG
Replies: 2   Last Post: Jan 3, 2000 10:38 AM

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 JIMMY Posts: 7 Registered: 12/6/04
T, TR, TV, TG, THG, TRVG, and TRHVG
Posted: Jan 1, 2000 3:40 PM

Do you know what to call the 7 possible frieze patterns?? In the title
of this message, it tells you what they are called. T, R, H, V, and G
stand for translation, 180 degree rotation, horizontal line
reflection, vertical line reflection, and glide reflection. Thus, the
T frieze pattern means translation only, like this:

'' '' '' '' '' ''
' ' ' ' ' '
-------------------

TR means translation and 180 degree rotation, like this:

'' '' ''
' ' '
-----------------
' ' '
'' '' ''

TV means translation and vertical line reflection, like this:

'' '' '' '' '' ''
' ' ' ' ' '
-----------------

TG means translation and glide reflection, like this:

'' '' ''
' ' '
-----------------
' ' '
'' '' ''

THG means translation, horizontal line reflection, and glide
reflection, like this:

'' '' '' '' '' ''
' ' ' ' ' '
-----------------
' ' ' ' ' '
'' '' '' '' '' ''

TRVG means translation, 180 degree rotation, vertical line reflection,
and glide reflection, like this:

'' '' '' ''
' ' ' '
-----------------
' '
'' ''

TRHVG means translation, 180 degree rotation, horizontal line
reflection, vertical line reflection, and glide reflection, like this:

'' '' '' '' '' ''
' ' ' ' ' '
-----------------
' ' ' ' ' '
'' '' '' '' '' ''

Now, look at all my examples of each frieze pattern and check to see
if each one is correct. Each one must have every one of the formations
mentioned in the name of the pattern and no others. Note that T, TR,
TV, TG, THG, TRVG, and TRHVG are the only possible frieze patterns.
You can prove there is no TH, THV, or TRV as follows:

Try to make a TH pattern with the figure I used to make my examples.
There should be no 180 degree rotation, vertical line reflection, or
glide reflection. But look carefully and you will see that there IS
glide reflection and your pattern is THG. Try to make a THV. There
should be no 180 degree rotation or glide reflection. But look
carefully and you will see that it has ALL 5 formations and your
pattern is TRHVG. Try to make a TRV. There should be no horizontal
line reflection or glide reflection. But look carefully and you will
see that there IS glide reflection and your pattern is TRVG. From
this, we know there is no TH, THV, or TRV.

Date Subject Author
1/1/00 JIMMY
1/3/00 Heidi Burgiel
1/3/00 John Conway