Difference between revisions of "Identifying Frieze Patterns Exploration"

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==Exploration==
 
==Exploration==
===Notation===
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===Crystallographic Notation===
You may use the notation in the text, or use this notation:
+
 
 +
The crystallographic notation for the frieze patterns is made up of four letters/numbers. The label always starts with a P, and the rest of the label is determined by the symmetries.
 +
 
 +
{| border="1"
 +
| P
 +
| M or 1
 +
| M, a or 1
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| 2 or 1
 +
|-
 +
| The first symbol is always a P
 +
| Vertical mirror line gives and M, <br>  otherwise we have 1
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| If the axis is a mirror line we get M, <br> if there is a glide reflection we get a, <br> otherwise we have a 1
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| Two fold rotation gives a 2, <br> otherwise we get a 1
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|}
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===Alternative Notation===
  
 
The border patterns can be given fairly simple names consisting of 2 symbols.
 
The border patterns can be given fairly simple names consisting of 2 symbols.
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(This notational system is derived from these meanings: M designates a mirror symmetry, G a glide-reflection symmetry, 2 a 2-fold rotation symmetry, and 1 the absence of a symmetry.)
 
(This notational system is derived from these meanings: M designates a mirror symmetry, G a glide-reflection symmetry, 2 a 2-fold rotation symmetry, and 1 the absence of a symmetry.)
  
Determining the symmetry group can then be accomplished by following this set of questions (we assume that the border patterns run from the left to the right, so that the terms horizontal and vertical are unambiguous):
+
Determining the symmetry group can then be accomplished by following a set of questions (we assume that the border patterns run from the left to the right, so that the terms horizontal and vertical are unambiguous). The order of the questions is important!
 
 
I (first symbol). Are there vertical lines of reflection?
 
 
 
:Yes -> the symmetry group starts with M.
 
 
 
:No -> the symmetry group starts with 1.
 
 
 
II (second symbol). Is there a horizontal line of reflection?
 
 
 
:Yes -> the symmetry group ends with M.
 
 
 
:No -> next question
 
 
 
::Is there a glide-reflection symmetry?
 
 
 
:::Yes -> the symmetry group ends with G.
 
 
 
:::No -> next question
 
 
 
::::Is there 2-fold rotational symmetry?
 
 
 
:::::Yes -> the symmetry group ends with 2.
 
 
 
:::::No -> the symmetry group ends with 1.
 
 
 
  
 +
{| border="1"
 +
| M or 1
 +
| M, G, 2 or 1
 +
|-
 +
| Vertical mirror line gives and M, <br>  otherwise we have 1
 +
| Is the axis is a mirror line? - M; <br> Is there is a glide reflection? - G; <br>  Is there a 2-fold rotation? - 2; <br>  otherwise we get a 1
 +
|}
  
 
The correspondence between this system and the one in the text (the IUC notation) is as follows:
 
The correspondence between this system and the one in the text (the IUC notation) is as follows:

Revision as of 15:03, 1 September 2015


Time-40.svg

Objective: Become familiar with identifying border patterns as well as creating them.

Materials

Printer.svg Printed version of the Identifying Frieze Patterns Exploration, dimmed: File:IdentifyFrieze.pdf
  • Printed copy of the Identifying Frieze Patterns Exploration.

Exploration

Crystallographic Notation

The crystallographic notation for the frieze patterns is made up of four letters/numbers. The label always starts with a P, and the rest of the label is determined by the symmetries.

P M or 1 M, a or 1 2 or 1
The first symbol is always a P Vertical mirror line gives and M,
otherwise we have 1
If the axis is a mirror line we get M,
if there is a glide reflection we get a,
otherwise we have a 1
Two fold rotation gives a 2,
otherwise we get a 1

Alternative Notation

The border patterns can be given fairly simple names consisting of 2 symbols. The first symbol is either 'M' or '1', depending on if the border pattern has a vertical line of symmetry. The second symbol is 'M', 'G', '2' or '1', depending on what other symmetries are present. (This notational system is derived from these meanings: M designates a mirror symmetry, G a glide-reflection symmetry, 2 a 2-fold rotation symmetry, and 1 the absence of a symmetry.)

Determining the symmetry group can then be accomplished by following a set of questions (we assume that the border patterns run from the left to the right, so that the terms horizontal and vertical are unambiguous). The order of the questions is important!

M or 1 M, G, 2 or 1
Vertical mirror line gives and M,
otherwise we have 1
Is the axis is a mirror line? - M;
Is there is a glide reflection? - G;
Is there a 2-fold rotation? - 2;
otherwise we get a 1

The correspondence between this system and the one in the text (the IUC notation) is as follows:

  • 1M = p1m1
  • 1G = p1a1
  • 12 = p112
  • 11 = p111
  • MM = pmm2
  • MG = pma2
  • M1 = pm11

Questions

  1. What is the symmetry group for the following border pattern: ... FFFFFFFFFFFFFFFFFFFFFFF...
  2. You can form all 7 border patterns if you start with F. Show the other 6.
  3. What is the symmetry group for the following border pattern: ... BBBBBBBBBBBBBBBBBBBB...
  4. You can form all 7 border patterns if you start with B. Show the other 6.
  5. What is the symmetry group for the following border pattern: ... OOOOOOOOOOOOOOOOOO ...
  6. You can form all 7 border patterns if you start with O. Show the other 6.