Assignment 1

(100)

 

You can use the Geogebra Program to do this assignment. Click here to download it. Use the Download button unless you already have java installed on your computer. You should use the template here to enter your answers in. Just paste your images from Geogebra into the boxes.

 

You can also just use a compass and straightedge. You will probably want to copy your work at each stage.

 

Part I: Triangles.

 

  1. Draw a Vesica Pisces.

image

  1. Connect the two points where the circles overlap with a line. Connect the two centers of the circles with a line as well

image

  1. Complete the two equilateral triangles formed by the points. Extend the sides of the top triangle down to meet the circles.

image

 

  1. Complete the two new triangles formed by the point of intersection of these extended lines with the circles. You can erase all the lines except the large triangle now.

iamge

 

  1. Connect each midpoint with the vertex across from it.  Where have you seen this triangle figure before? It may help to erase the circles.

image


 

 

Part II. Squares.

 

  1. Start with a Vesica Pisces and draw a horizontal line connecting the midpoints. Then  draw a line perpendicular to the line at its two endpoints.
    II1-image
    1a. ( Compass and penicl users only): you will have to do the following construction ot get the prependicular lines.:Draw a circle with center B and radius BC. Draw another circle with center B and radius BD. Draw a circle with center D and radius DB. Draw a circles with center C and radius  CB and CA. We will be interested in the points where these overlap in the center. You need not finish the circles to the far right and left. This seems complex, but you are really  just constructing perpendicular lines at points B and C. BC will form the base of the square.


Mark the points where the perpendicular lines meet the two circle at the top and connect the four points to complete the square. you can hide the perpendicular lines. 


 II-1


 

3. Mark the midpoints if the bottom and top of the square, using the center line of the vesica pisces. You may now erase everything but the square. Save at this point, or make a copy. You will need this again later.

Find the midpoints of the two sides of the square. You will have to use the line bisector in Geogebra. If doing it by hand you can measure to find an approximate midpoint, or, better, use these instructions to bisect the lines manually.

Now connect the diagonal points of the square. Make a smaller square by connecting the four  midpoints of the lines that make the original square.  Now connect the diagonals of this smaller square. Where have you seen this diagram before?

image


 

4. Make a smaller square inside the second square by connecting its midpoints (marked by the diagonals of the original square) in the same manner as above.  Now make a fourth larger square outside the original square. Extend the two midpoint lines of the original square, and construct a line at the top left corner that is parallel to the diagonal until that line meets the extended midpoint lines. Repeat for the other three corners.

image

 
 

 How is each square related to the diagonal of the next smaller square? What are the relationships between  the sizes of the four squares?

5. 

Start with a square again, from step 3 above. Draw the diagonal from point A to D. Draw the lines perpendicular to that diagonal AD at points A and D. Then draw a Vesica Pisces with a circle center A, radius AD and a circle center D, radius DA.

II-5

5a..

Mark where the circles intersect the perpendicular lines and complete the square built on the diagonal AD, by making the segments DF, FE, and EA.

II-5a

6.  Now make another diagonal  CB in the original square. and repeat all the steps from number 5 for this diagonal to create another square based upon it, CBHG. 

II-6

How many squares do you see?

Indicate the relationship between the areas and the sides of the different size squares.

Squareareaside
Unit square ABDC 11
larger square
smaller square

 

 

III.             The Golden Section and the Pentagon:

 

1. Construct a Golden rectangle.

Start with a square. Use the regular polygon tool or start from a copy of the square constructed above.

Bisect the bottom of the square and then continue that bottom segment in both directions.

Draw a circle with center E and radius ED to intersect the bottom line. You are inscribing the square in a semi-circle. Mark the point where the circle hits the line F. The line AF is cut by B in the golden section.

drawing

Mark the other point where the circle hits the bottom line G. Extend CD in both directions. Raise perpendiculars up at F and G.  Mark the two points where these hit line CD,  H and I. GHIF is a Square Root of 5 rectangle and ACIF and GHDB are Golden Rectangles.

 


2. The Golden Spiral.

Start with a Golden rectangle ACIF above. Note that BDIF is also a golden rectangle.

 

Measure out on DB and IF  a length equal to DI. Mark these points J and K. Make the square JDIK

BJKF is also a Golden Rectangle.

 

Repeat the procedure a couple more times.

 

Draw an arc from A to D with radius BA. Then do the same with the next smaller square: an arc from D to K with radius JD.

Continue with the next smaller square and so on as far down as you can get. This is the Logarithmic or golden spiral.


3. Pentagon:

  Start with a line divided in a Golden section, such as ABF from above. You can also reconstruct one using the square root of 5 rectangle method from above.


 

Draw circle with center A and radius AB and another circle with center B and radius BA.

 

Now draw a circle with center A and radius AF. Then another circle with center B with the same radius AF. (You will have to measure AF and use the circle with determined radius function in Geogebra) Mark the points where the two large circles intersect each other and the two small circles. Connect each of these points with each other and with AB to make the pentagon.


 

4. Pentagram Star.

Start with the pentagon. You may erase all the guidelines. Connect each vertex with the one directly opposite it. This will give you a pentagram star inside the pentagon.

You can repeat this process again within the internal pentagon.
Extend each of the sides of the original pentagon to make a larger pentagram outside.

 

How many instances of the golden relationship can you find between the parts of the pentagram?

 

 

IV. The Platonic Solids:

Start with a vesica pisces divided into 4 triangles as  in part I above:

image
Remove the circles and fold you have a tetrahedron.

 

Extend the vesica pisces to six circles and use it to trace out these 6 squares:

 

 

The same pattern with 5 circles will give the octahedron:

 

 

Use the  same pattern with 8 circles for the icosahedron:

 

Just hand in the drawn or printed templates.

 


Extra Credit ONLY

You can cut them out and actually construct the solids for extra credit. If you like , you can print out these already drawn templates to make your models.

Tetrahedron                          Cube                       Octahedron           Dodecahedron                      Icosahedron

 


PART V: Finding the Golden Section

There are a number of ways you can do the measurements in this section: (the three kinds of measurements below in questions  1, 2, and 3.)
1. Use a ruler and calclulate the ratio of sides to see how close it comes to .618 or 1.618. This will be the least exact method.
2. Print out these templates: vertical and horizontal in different sizes or use them on the screen and use them to measure objects.
Here is another template.
3. Use one of these programs.  Both of these are shareware. You can use then free for 30 days.
They produce a template that appears transparently over your screen you can move around and resize to see if things are in the golden proportion.
Phi Matrix is simpler to use. Atrise has more features.
Phi Matrix http://www.phimatrix.com/PhiMatrix.exe

Atrise Golden Section   http://www.atrise.com/golden-section/ags.exe

Required Measurements. You must do the minimum. You can get extra points by doing more and finding really interesting things. You will get less credit if you go to a site that already has collected images in the golden section for you. Try to find your own. 

1. Measure at least 5 commonly used man made objects. Can you find at least one that has proportions in the golden section?

Object

Feature

Golden Section?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2. Look at some pictures of art works or architecture. Find at least 5 that have prominent features in the golden section.List their names and the features measured. You might also find this template useful in looking at larger objects. You may find this table helpful in reporting your results

Name Feature Artist (if known) Web address (if from web) or source Image
(optional)
         
         
         
         
         

3. Look at some natural objects and parts of natural objects (any living thing, part of a living thing, or highly organized inorganic object, such as a crystal. Don't use anything that has been reshaped by Man, such a cut gem or carved wood.) Find and list at least 8 examples (at least 4 should be from different objects, not parts of the same object) of the golden section. Did you find any organized natural objects whose main divisions were not in the golden proportion? List them. You may find this table helpful in reporting your results

Object Part in golden proportion Image or drawing (optional) web address:(if from web) or source
       
       
       
       
       
       
       
       


2006 David Banach 

Creative Commons License This work is licensed under a Creative Commons License.

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