(Updated Jan. 7, 2015) We finally published “Green Math: How to Apply Geometry to a Square Foot Garden” in Decoded Plants. Perhaps I was right to highlight these images. The editor rejected my adaptation of Toby Hudson’s circles, but hand-crafted some original art in its place. As well, fellow writer Chris Eirschele kindly offered several images of actual container gardens to show what one can do within a limited plot of land. Finally, I didn’t even include the “square root of 2” image, since it didn’t really add that much for a gardening perspective.
Here’s a preview of images I made for an upcoming article about applied geometry. Now that my article is published, I’ve shared the link and further details. (End of update).
Why preview these images? The act of publishing them first on my own online site establishes copyright.
Packing Many Circles into a Square
- “Pack Circles into Squares”. Copyright image by Mike DeHaan based on original work by Toby Hudson @ http://commons.wikimedia.org/wiki/File:Circles_packed_in_square_2.svg .
Most of the work was done by Toby Hudson, who published it online under a Creative Commons license. My work was to capture a screen-print of all these images, and then crop the image and add a border. Hopefully the finished article will show it as a larger image so it will be easier to read.
Perhaps it’s obvious that my article will focus on how to pack circles into a unit square as efficiently as possible.
Pythagorean Theorem for an Isosceles Triangle
- “Square Root of Two in Square Foot Garden by Pythagorean Theorem” image by Mike DeHaan
I drew this image in MS Paint to illustrate a 1-1-root(2) triangle. Per the Pythagorean Theorem, the length of the longest side is the square root of the sum of the squares of the other two sides.
Since the outside edges of the square have a length of “1 unit”, the diagonal of the square is the hypoteneuse of a 1-1-√2 right-angle isosceles triangle.
The Pythagorean Theorem says that hyp^2 = side_one^2 + side_two^2. Since each side has length of 1, hyp^2 = 1 + 1 = 2. So the length of the hypoteneuse = the square root of 2 = √2, or about 1.4.
Red Circles Do Not Quite Fit Inside the Square
- “3 Circles Too Large for a Unit Square”. Copyright image by Mike DeHaan.
I wanted to fit three circles along the diagonal of this square. They fit along the diagonal, but bulge out beyond the sides of the square.
Obviously the circles did not fit perfectly along the diagonal either. I had converted the unit square from feet to inches, because the article is mainly aimed at readers in the USA. The square root calculation rounded more closely to 5.6 inches, so I rounded down to 5½. The red circles would have bulged out even more had I been more careful with the fraction-of-an-inch!
I used MS Paint to crop these images and add a border. At first, I’d forgotten to doctor the image after moving it from the camera to my computer. I was surprised that the original image of the green circles was “too large” to be uploaded to this site; but it had accepted the original red circle image.
I could have changed the WordPress parameters for my site, to allow larger images. However, I would have cropped them both anyway. There was just no need to include so much of the “landscape” that the camera captured.
Green Circles Fit the Square
These green circles do indeed fit inside the square. Unfortunately, they don’t reach to the corners of the square.
I wonder whether that will become a sore point with the readers who want to apply this geometry to their practical situation. Perhaps there will be feedback for that article.
Learn from the Father of Square Foot Gardening
Mel Bartholomew has not just “written the book”, he’s written several books on Square Foot Gardening. You should read the Kindle edition of “All New Square Foot Gardening, Second Edition: The Revolutionary Way to Grow More In Less Space“, available from Amazon in Canada and from Amazon in the USA.
Professional Writing by DeHaan Services
DeHaan Services writes web site or advertising copy for clients. Our skills are demonstrated in online articles in Decoded Science, and other Decoded sites. That’s why we highlight these online articles on this blog page.
I also publicize many of my articles in another blog; read “Geometry Applied to Square Foot Gardening” for a writing tip.
Thank you for reading this preview of applied geometry.