Last Update 11/ 07/ 2005
in English / in Esperanto/ in Portuguese
A click on any of the first 7 buttons on the right side of the figure counted from top to bottom will be able to start the execution of selected symmetry operations on a given straight line plotted in the cartesian plane. The symmetry operations are performed by the symmetry elements or operators described in table 1.
Table 1. Description and orientation of the selected symmetry elements.
| Symbol | Description | Orientation; contains the origin and |
| m | mirror plane | perpendicular to x |
| m | mirror plane | perpendicular to y |
| 2 | binary axis | perpendicular to xy plane |
| 3 | triad axis | perpendicular to xy plane |
| 4 | tetrad axis | perpendicular to xy plane |
| 6 | six fold axis | perpendicular to xy plane |
| 2m | binary axis contained
in the mirror plane |
binary axis perpendicular to xy plane
and m perpendicular to x or y |
A click near to the original straight line (in blue) will show its general
equation, the coordinates of the head and tail of the segment and the equation
in the slope intercept form (also in blue).
Repeated clicks on buttons 3 or
4
or
6
or 2m will generate the straight lines of the answer set for that
respective symmetry element. Rotation axes 2,
3,
4
and
6
are
cyclic operators, it means that repetition of previous results will occur
after a turn of 2p radians, as explained here.
The symbol 2m represents a group of two symmetry elements, as can
be seen in table1.
Equivalent points
The coordinates of equivalent points resulting by the action of the symmetry elements in this page up on the coordinates of the initial point are organized in table 2. The orientation of the symmetry elements in table 2 is given in table 1.
Table 2. Analytical equations for the coordinates of equivalent points
| Symmetry element | Initial point | Equivalent point |
| m | [ x ; y] | [ (-x) ; y] |
| m | [ x ; y] | [x ; (-y)] |
| axis, a = p, p/2, 3p/2, 2p/3, 4p/3, p/3,... | [ x ; y] | [(y.sina + x.cosa) ; (y.cosa - x.sina)] |
The clockwise rotation was adopted for the axes in table 2.
Exercises
1) Pulg the coordinates of the head and tail points of the original blue segment into the proper analytical equation of symmetry and calculate the coordinates of the ends of each segment in the respective answer set.
2) Calculate for each case of the previous exercise the general form equation and the slope intercept equation for the straight line.
3) Why for each of the operators m, m, and 2 in a given orientation only one answer is generated when applied to the original straight line?
4) Compare the slope of perpendicular straight lines. What symmetry elements are able to generate perpendicular straight lines?
5) Compare the slope of parallel straight lines. What symmetry elements are able to generate parallel straight lines?
6) What can be observed about the x and y independent term of a straight line equation in the general form and in the slope intercept form after comparison of these values for the original straight line and for the other one, obtained after application of the 2 operator?
7) How can you justify the intersection of the original straight line and the other obtained by mto occur on the y axis?
8) What other operator or group of operators used in this study will give at least in one of their answer the same straight line equation as the obtained by the symmetry element 2 operated on the same original straight line?
9)What is the condition to have the same equation for the original straight line and for the straight line obtained by application of the binary axis on the former?
10) What interesting information is obtained by calculating the angle related to the slope of the original straight line and the slope of the next straight lines obtained by consecutive actions of any selected rotation axis?
11) How can you explain that coordinates for ending points of straight line segments obtained by some symmetry elements are always integers and when obtained by other symmetry elements may be not integers?
12) Consider the statement: The symbol 2m means the same as 2mm or a binary axis contained in the intersection of two perpendicular mirror planes. Is this true or false?
A click on button C will show the answers for the action of that selected symmetry element.
Please send your comments.
Table of subjects.