How to Make Expanding Golden Rectangles

We have been getting a tremendous amount of inspiration from the book A Beginner's Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art, and Science by Michael S. Schneider as we work through the quality of numbers 1-10. We have rested on the pentad for several lessons spanning weeks as this lengthy chapter is rich with information and examples. While we have previously done the Golden Triangle with the pentagon and pentagram specifically, we have moved to the Golden Rectangle and worked with nesting and expanding rectangles as once you make the first, it's a simple process to make them 'grow' or 'nest'. It may be simple to construct a Golden Rectangle, but the types of tools you use to construct it will play a large role in the accuracy of your final result. After many trials and errors, I found that using a pencil and ruler was not as precise as using a mechanical pencil, calipers and a straightedge. While you may be able to get one Golden Triangle with fair accuracy with a ruler and pencil, your accuracy will increase with a mechanical pencil (or very sharp pencil) and calipers. If you wish to make nesting Golden Rectangles, your tools will be crucial as I was only able to get a max of three nesting Golden Rectangles before the measurements were so distorted, it looked like either a square and skinny rectangle or two squares.

To construct a Golden Rectangle, you may begin in one of two ways (and there are many other ways). You can simply start with the corner of your paper as that it already a right angle. I recommend measuring both sides of your paper at your desired measurement (3-4 inches is sufficient), and drawing a line across your paper. Then measure 3 or 4 inches (you are making a square, so the measurements must be consistent), and draw a line you you've created a square. You will have an extra line across the page, but don't erase it just yet.

Next, you will need to bisect your square. I used calipers or a ruler (calipers are best), and found the midpoint. Mark it on parallel sides of your square. Next, take your compass needle and set it at the midpoint and extend the compass so it reaches the opposite corner of your square. Swing an arc up and repeat with the other side of the square. Now, connect the point where the line and the arc cross with the otherside of the arc (if you worked at the edge of your paper, it's where the arc crosses the edge of your paper). Now you've created your very own Golden Rectangle.

Slow step by step tutorial on how to construct a Golden Rectangle
   • How to Make a Golden Rectangle  

Fye or Phi (ϕ) is a fascinating number that exhibits a ratio between two lengths. The Golden Proportion is the smaller portion to the larger portion is equal to the larger portion to the whole. The Greater Golden number is 1.6180339 and the numbers go on to infinity. There are no whole number fractions that equal the Golden Number. The lesser Golden number is 0.6180339 and the number goes to infinity as well. To get the lesser Golden number, measure the short part of your rectangle and divide it by the long part and find your ratio!

To make expanding Golden Rectangles, draw straight lines from the long side of your rectangle using a straightedge. Then mark them with the measurement of your long side to create a square attached to your original Golden Rectangle. Connect the points and erase the extra line and now you can keep going to make even more Golden Rectangles! You make need to add more paper as the expanding Rectangles get big fast.

Check out the blog post that accompanies this video for more information
https://www.pepperandpine.com/waldorf...

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