It began with mention of the Golden Ratio on some of the Engineering an Empire videos we were watching. Then we read a little chapter about the golden rectangle in Penrose the Mathematical Cat and we found some other you-tube videos, particularily this one about using a golden ratio gauge in designing woodworking projects.
My first searches for instructions on making a golden ratio gauge all turned up the same proportions: 34, 21 and 13 centimeters. I made one like that and then decided to try scaling it down. Dividing all the distances in half worked. Armed with two golden-ratio gauges of differing sizes we did a golden-ratio hunt searching out objects in the house that were built according to the golden ratio. There’s actually a freakishly large number of them so it was quite fun to do.
I was curious about the measurements of the gauges. Could I make one without a ruler? The length of the pieces are all a golden ratio to each other so it doesn’t matter what size you start with. Here’s the step by step instructions I developed:
1) Draw a square in the corner of a piece of cardpaper. I use a second piece of cardpaper as my straightedge so that I can mark on it how far the sides of the first line is. If you’re making a small gauge for drawing with then about an inch and a half is a good length. If you are making a big gauge you might need to use posterboard.
2) The next step is to expand the square outwards in order to make a golden rectangle. To do this, find the center of one edge of the square. If you are using cardpaper as a straightedge then you can do this easily by folding the cardpaper to find the middle. Mark it on your drawing and then draw a diagonal line between that spot and one corner of your square. Mark this distance on your straightedge or adjust a compass so that you can draw an arc with this radius down towards the bottom of the paper. The radius of this arc is the distance between the middle of one side of the square and the outer edge of the golden rectangle. So draw the rectangle.
3. Now expand the rectangle upward in order to make a square.
4. Repeat for this square what you did to the first one. Find the middle of one side, measure from there to a corner and then use that as the distance between the middle spot and the edge of the rectangle. Your picture should look something like this:
What you have is a set of golden rectangles. Can you find all of them? There are two “tall” ones and then two “long” ones that are made up of the tall ones plus the square next to them.
The point of all of this is to get the three measurements you will need. These are the three vertical lines in my picture. Use your straightedge and extend them all the way across the page. Then small square is just about the size of the smallest piece of the gauge. The golden rectangle made from that square is about the size of the next piece of the gauge and the distance to the outer part of the biggest rectangle is about the size of the longest parts of the gauge.
I say “about the size” because we need to put holes in them in order to attach them together and the holes can’t be at the edge of the gauge, so what I do next is add a centimeter to my pieces. The smallest piece will need two centimeters added because it has holes at both the ends. The other pieces only need one centimeter added. At this point my sheet looks like this:
Once you have the strips measured off you can cut them. You need two longer strips, one middle sized and one short one. Punch holes one centimeter in from the edges and use fasteners to attach the pieces together as in the top picture. You can carefully cut the tips into points and explore your home for golden rectangles.