Assignment 1

Due 2/6

(100)

 

You can use the Euklid Program to do this assignment. Click here to download it.

 

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.

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

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

 

  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.

 

  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.

Here is the Euklid file for this construction. You Load it then hit Miscellaneous then Replay  (CNTRL O R) to see the construction step by step.

 

 

Part II. Squares.

 

  1. Start with a Vesica Pisces and draw a horizontal line connecting the midpoints. Label  the 4 points where the line hits the circles A-D from right to left.
  2. Draw a circle with center A and radius AC. Draw another circle with center B and radius BD. Draw a circle with center D and radius DB. Draw a circle with center C and radius 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 circles intersect on the vertical lines at points B and C. Connect the points below and above to make the sides of the square. These lines are complete in the diagram above.

Mark the points where these vertical lines intersect the two original circles of the vesica pisces. These are the two top corners of the square. You can now complete the square on base BC.

 

Here is the Euklid figure that you can replay to see the construction.

 

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.  Find the midpoints of the two sides of the square. You will have to use the line bisector in Euklid. 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?

Here is the Euklid file.

 

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 (use the parallel function in Euklid) until that line meets the extended midpoint lines. Repeat for the other three corners.

 

Here is the Euklid file .

 

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

 

 

III.             The Golden Section and the Pentagon:

 

1. Construct a Golden rectangle.

Start with a square. Load this file, 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.

 

Here is the Euklid file.

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.

Here is the Euklid file.


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.

For extra credit you can try this construction of the golden section.

 

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 Euklid) 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.

Here is the Euklid file.

 

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?

 

For extra credit you can try this construction of the pentagon from the vesica pisces.

 

 

IV. The Platonic Solids:

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


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 patter with 8 circles for the icosahedron:

 

Just hand in the drawn or printed templates.

 

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: Building a Golden Section Gauge

A. Making the gauge.

1. You will need two straight thin objects from 10 to 20" long that can be punctured or drilled in the middle. Longer is better. A long stick like a kabob stick or chop stick is best, but in a pinch a long sturdy plastic straw will do.

2. You will find the golden section point of both sticks and mark it. You can mark the length of the stick on a piece of paper and then divide that line according to the golden section using the methods Here or Here. (The second is the simplest). Then once you find the golden section point on the line, you can mark the spot on your sticks.

You could also measure the stick and calculate the length .6180 of the original length and then measure and mark that length. (This is less accurate, but works well enough for the degree of precision you can achieve with these materials. Measuring and marking very carefully will make a difference.

You could also try drawing a line the same length as your stick in Euklid, dividing it according to one of the methods above,and then either using the measurements or printing it out as a template to mark the stick. If you print you will have to be careful to print the actual size.

3. Make a hole in each stick at the point you marked. You can make this with a pin, a thumbtack, a nail or a drill. Please do not poke, puncture, impale, or otherwise injure yourself. See me if you can't figure out how to do this, or ask an adult for help ;)

4. You will need to attach the two sticks a the holes with a pin, nail, paperclip, or rivet. Make sure that the sticks move, but are not loose. You will want the sticks to hold their position and not swing freely. You should make the hole somewhat smaller than the object you will use to hold them together so the fit is snug. (For example, make the hole in a straw with a pin, but use a paper clip to hold them together.) Make sure to trim any sharp or dangerous ends.

5. Trim the ends of your sticks to make sure they are the same length. You may also trim the ends of straws to points or sharpen sticks in a pencil sharpener to make them more precise pointers. 

6. You are done. No matter how you move the sticks the two ends will measure out lengths that are in the golden proportion. You must hand in your gauge with the assignment in a plastic bag with your name on it.

B. Testing your precision.

Open the larger end to 6, 12, 20 and 30 cm. and then measure the length in cm ( to at least a tenth of a centimeter precision) of the smaller end of the gauge. If you can't open to 30, try 25.

Copy this table onto your assignment and fill it in with your data.

  Long end Short end Ratio Percent off
1 6.0 cm.      
2 12.0 cm      
3 20.0 cm      
4 30.0 cm      

Your ratio would be long end divided by short end.

Your percent off would be [(1.6180- your ratio) / 1.6180] x 100.

You may want to add a short explanation of your precision. Points will be awarded for really good precision and taken off for really bad precision. Bad Karma will result from lying about your precision.

C. Measurements. You must do the minimum. You can get extra points by doing more and finding really interesting things.

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


© 2006 David Banach 

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

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