Homothety, Rotation and Symmetry in Arts

Homothety, Symmetry and Rotation in Arts Worksheet for using GeoGebra with examples and applications 9C Grade “Jean Monnet” High School June 2018 Teacher: Mihaela Gîț

Homothety & Symmetry in Arts - Richard Anuszkiewicz Richard Anuszkiewicz (born May 23, 1930) is an American painter, printmaker, and sculptor

https:// a.1stdibscdn.com/archivesE/upload/a_466/1492097428788/A nuszkiewicz_Inward_Eye_3_master.JPG https:// encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcS8pVctB Ff5sPfdpSfi7kcmrl2NcqkGeLyPX0nXhJTzpG59GaRW_Q

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The use of homothety and symmertry of figures in Geogebra, for Anuszkiewicz ‘s paintings • Open a file GeoGebra and save as Name_Surname (or a suggestive name) in File Menu.

• Draw a square ABCD –Option “Regular Polygon” (Icon 5).

• Draw the triangle BCD –Option “Polygon” (Icon5).

• Draw the segment [AC] – Option “Segment” (Icon 3).

• Draw the point on the segment [AC] – Option “Point” (Icon2); the notation will be automatically E (it can be changed –right click on point, choose “Rename” and change the letter &OK).

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Draw the intersect point F, between AC and BD –Option “Intersect” (Icon 2).

Draw ΔB’C’D’ similar with ΔBCD, using Option “Dilate from Point” (Icon 9)- with center E and factor .

Right click on the triangle B'C'D‘ to change its properties: color, line thickness, etc.

Move (Icon 1) the point E, for a better position of the triangle B’C’D’ on AC. To create an animation, right-click on E and tick "Animation On“.

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Repeat the process at least 3-4 times; color each time the triangle obtained.

Be careful to select the desired triangle!

The figure can also be selected from the "Algebra“ window (Select “Algebra” in View Menu).

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Draw the triangle with ΔBCD relative to point F. Repeat the process for each triangle.

Attention! It is good to know that the figures are drawn on the layers; the last figure is above the others; if necessary, you can change the order of layers using the way: Properties window→Advanced →Layer →choose the number of the layer and close the Properties window.

For beginners, it is recommended to work with transparent figures to see how its are 6 layered.

• To hide the labels: Select the Move button, right-click and select all drawing elements in a rectangle and then right-click - properties - to uncheck Show Label.

• Export the draw as a picture: Bar Menu File → Export →Export as Picture (.png) in the selected folder.

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Homothety & Symmetry - Task Using the homothety and symmetry properties, draw new paintings in GeoGebra, so that: 1. the first shape is a square with: l=4u and k=

2. the first shape is a quadrilateral; 3. the first shape is a regular hexagon.

Victor Vasarely - original inspire us to work in GeoGebra https:// thumbs.dreamstime.com/z /vasarely-optical-effec t-26404683.jpg 8

Homothety & Rotation in Arts – Victor Vasarely 1906– 1997, was a Hungarian-French artist, who is widely accepted as a "grandfather" and leader of the op art movement.

The paintings of these links can be built using GeoGebra tools, using successive homothetics and rotations.

https:// image.shutterstock.com/image-vector/black-white-optical-illusion-v asarely-260nw-529335223.jpg https://previews.123rf.com/images/elenasz/elenasz1301/elenasz130100009/1734482 -black-and-white-optical-illusion-vasarely-optical-effect-. jpg 9

The use of homothety and rotation of figures in Geogebra, for Vasarely ‘s paintings 1. Open a file GeoGebra and save as: Name_Surname (or a suggestive name) in File Menu.

2. Draw a square ABCD – Option “Regular Polygon”

(Icon 5).

3. Draw the center E of the square ABCD – Option “Midpoint or Center”

(Icon 2)

4. Draw the homothetized square with ratio k=5/6 from point E – Option “Dilate from point”

(Icon 9).

5. Select the option “Rotate around Point”

to draw the rotated square by, around E

(clockwise, for example).

6. Hides the previous homothetized square (from point 4) - right-click on the drawing and uncheck the “Show Object “ option or in the Algebra window uncheck the blue dot next to the polygon.

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Right-click on the drawing (or the figure label in the Algebra window) to use coloring / transparency properties.

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After 5 stages (steps 4-5-6), you get the next drawing:

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After 10 stages (steps 4-5-6), you get the next drawing:

• Mathematical Tasks for -Calculates the ratio of the perimeter of two consecutive squares.

-Calculates the sum of all the squares perimeters.

-Calculates the ratio of two consecutive squares in this drawing.

-Calculates the ratio between the smallest square area and the largest square area in this drawing. How many squares I need to build, to get the ratio ?

-Calculates the sum of all the squares areas.

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9C – Works Homothety-Symmetry

Alex C.

Sofia M. 12

9C – Works Homothety-Symmetry

George B.

Theo I. 13

9C – Works Homothety-Symmetry

Maria S.

Ovidiu 14

9C – Works Homothety-Symmetry

Mara S.

Mihai 15

9C – Works Homothety-Symmetry

Mario

Alex T. 16

9C – Works Homothety-Symmetry

Vlad

George V. 17

Homothety-Symmetry – 2 Translations

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Homothety with and center E (center of the square) for each square Rotation for each square with angle around E

Cătălin I.

Sofia M. 19

9C – Works Homothety-Rotation

Mihai

Ovidiu 20

9C – Works Homothety-Rotation

George V.

Sofia S. 21

9C – Works Homothety-Rotation and animation

Alex C. 22

9C – Works Homothety-Rotation

Maria S.

Alex T. 23

9C – Works Homothety-Rotation

Alex D.

Mary 24

9C – Works Homothety-Rotation -Translation

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“Jean Monnet” High School June 2018 30