Symmetries

A triangle is rotated four times, then everything is reflected in the y-axis.
Everything is translated along the vector.

There are three symmetries in the plane: reflection symmetry, rotation symmetry, and translation symmetry. If two geometrical objects are such that one of the objects can be transformed to the other, by one of the three symmetries, then the objects have the same shape and the same size; such objects are called congruent objects.

A comparison between reflection and rotation symmetry is shown on the page Geometry - Tessellations and Symmetries.


Reflection symmetry

Exercise 1 - Draw a mirror image

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  • Make sure that the axes are not shown.
  • Enter a line between the points A and B.
  • Enter a free point C.
  • Use the tool Reflect Object in Line icon; click on the point C and then on the line. The mirror point C' is created.
  • In order to distinguish between the free point C (the point you can drag) and the dependent point C', you can change the look of the points. Make sure that the two points don't look the same.
  • Put a trace on both points by right-clicking on them and checking Trace On. Draw a picture by dragging the point C.
  • You can erase the picture drawn by zooming in or out.

Exercise 2- Reflect in two lines

Make another line perpendicular to the original line. Reflect both the point C and the point C' in the new line. Put a trace on all points and draw!

Exercise 3 - Reflect a polygon in a line

Enter a line and a polygon. Make a mirror image of the polygon.


Translation Symmetry

An example of several translated copies is shown on the page Linear Algebra - Translation along a Vector.

 

Dynamic tesselation made of points placed on triangles.
Drag the points!

A more advanced Processing-example of a tessellation is shown on the page Geometry - Tessellations and Symmetries.

Exercise 4 - Translate a polygon

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  • Make sure that the axes are not shown.
  • Make a vector u between two points A and B (use the tool Vector between Two Pointsimage).
  • Make a polygon.
  • Use the tool Translate Object by Vector image. Click on the polygon and then on the vector u. A new polygon is created.
  • Drag the points A and B to change the direction and the length of the vector, the position of the new polygon changes.
  • Drag the vertices of the first polygon, the new polygon changes. Note that you can't drag the vertices of the new polygon, these points are dependent objects.
  • Make several translated copies of the polygon by using the tool Translate Object by Vectorimage on the second polygon then the vector, on the third polygon and then the vector, ...

Exercise 5 - Make a tessellation

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  • Make a triangle (not a regular triangle).
  • Make a vector u through two of the vertices and another vector v through two other vertices.
  • Use the tool Translate Object by Vector image to make one copy of the triangle along the vector u and another copy along the vector v.
  • Make several copies of the new triangles.
  • Drag the vertices of the first triangle, the entire tessellation changes.
  • If you want to colour the areas between the triangles, you can make a new triangle through existing points that covers a white area; then make translated copies of the new triangle along the vectors u and v.
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Rotation Symmetry

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Exercise 6 - Rotate a polygon

  • Make sure that the axes are not shown.
  • Make a polygon and a point.
  • Use the tool Rotate Object around Point by Angle image. Click on the polygon, then on the point, choose an angle in the pop-up window.

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  • Move the rotating point (the point E in the picture above) and change the original polygon.
  • You can also rotate the polygon around a vertex of the polygon.

Exercise 7 - Vary the rotation angle with a slider

  • Make a polygon and a free point (you can also use one of the vertices of the polygon as the rotation point).
  • When making a rotated copy of the polygon you must specify an angle. Instead of specifying the angle as a fixed number, you can use a slider; with a slider you can vary the value of a number. Find the tool Slider icon and click anywhere in the drawing pad. Change the value of the slider from Number to Angle, the name of the slider becomes α.

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  • Use the tool Rotate Object around Point by Angle image. Click on the polygon, then on the point, then choose α as the angle (you can find the Greek letters in the menu to the right)

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  • Change the rotation angle by using the slider.

Exercise 8 - Vary the rotation angle by dragging

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  • Make two segments having a mutual endpoint.
  • Use the tool Angle icon to measure the angle. If the points are labeled as in the picture to the right, click on B, A, C; in that order.
  • The value of the angle is shown. In the algebra view, you can see that its name is α. In the Styling Bar you can choose if you want to display the name or the value.
    The angle α will now be used to specify the rotation angle.

image

  • Make a polygon and a free point (you can also use one of the vertices of the polygon as the rotation point).
  • Use the tool Rotate Object around Point by Angle image. Click on the polygon, then on the point, then choose α as the angle.
  • Change the rotation angle by dragging the points making up the two segments.

Many transformations

Exercise 9 - Make many transformations by using the tools

In the topmost applet, there are four rotations, five reflections and ten translations. After rotating the successive copies of the triangle 72° around the origin, it is possible to reflect the five triangles in the y-axis by selecting all of them. Select the tool "Reflect Object in Line", drag with the mouse to select all triangles, then click on the y-axis.

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Translating ten triangles along a vector can be done in a similar way.

Exercise 10 - Make many transformations by using the spreadsheet

When using the spreadsheet, you must write commands instead of using the tools. When writing a command in the input bar, the code completion will help, press enter to choose the suggested command. One way to find out how to write a command for a given tool, is to use the tool in the toolbar, and then check out how the object is defined in the properties window.

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The commands for rotating around the origin, reflecting in a line, and translating along a vector, are:

Rotate[ <Object>, <Angle> ]
Reflect[ <Object>, <Line> ]
Translate[ <Object>, <Vector> ]

Create a triangle. To make transformations using the spreadsheet, start by renaming the three points and the triangle to A1, B1, C1, and D1, respectively. Then click on View->Spreadsheet to see them in the spreadsheet.

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Click on the small arrow to "Toggle Styling Bar" for the spreadsheet. Then click on "Show Input Bar".

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Select cell A2. Input Rotate[A1,72°] in the input bar and press Enter. Write Ctrl + o to write the degree-symbol.

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Drag the square in the lower right corner of A2 over the cells B2 and C2 to make relative copies.

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Make a relative copy of D1 to D2.

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Select the cells A2 to D2 and make relative copies down over three more rows.

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Input Reflect[A1,yAxis] into cell A6 and make relative copies over cells B6 and C6. Make a relative copy of D5 to D6.

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Make relative copies of row 6.

A more general example, using lists, is shown on the page GeoGebra Tutorial - Lists.

For more information about the spreadsheet, see Geogebra Tutorial - Spreadsheet.

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License

www.malinc.se