Mel Bartholomew popularized Square Foot Gardening with his simple and practical approach: Create squares in a 4 foot by 2 foot (or 4’x4′, or 8’x2′) rectangle, and cultivate different plants inside each square.

The largest plants may need one whole square, but 4, 9 or 16 smaller plants might share one square. You work the cultivated area from outside, to avoid compressing the soil. Even a novice gardener can divide and conquer a garden, one square foot at a time!

How can we apply geometry creatively to square foot gardening? First, how can we best arrange the plants inside one square? Second, what shapes can we make the garden plot?

## How to Arrange Plants Inside One Square

One goal is to provide the most space for each plant. Another would be to pack the most plants inside a single one-foot square with enough spacing for the plant to thrive.

Remember that plants do not have “square roots”; each needs a circular area of its own. In geometry, the question is how to pack the most circles of a given size into a square.

The usual pattern is to arrange 1, 4, 9 or 16 plants in a simple square formation. We call this “square packing.” This is the ideal pattern for a “square number” of circles: (1, 4, 9, 16, 25, …). Space the rows and columns equally inside the square.

See the photograph of circles in a square. Where there are 4 or 9 “square packed” circles, it takes 2 or 3 in a row to cross the square. So the diameter of each circle must be ½ or 1/3 of the square’s one-foot length. In inches, that’s 6″ or 4″ spacing per plant.

The diameters of the other circles are harder to calculate. When packing 2 or 5 circles into a square, the temptation is to apply the Pythagorean Theorem to calculate the diagonal of the square (about 1.4 feet), and then divide by the number of circles along the diagonal.

But from the diagram, it’s obvious that all of the circles do not touch the corners of the square. So there’s always wasted space.

The spacing between plants is the diameter of a circle drawn around that plant. How many circles of a given diameter fit into a 1-foot square? I used an online math resources page to find the first nine answers for a “unit square.” Then I converted to inches, and rounded down:

- For 12″ spacing, plant 1 in the center.
- For 7″ spacing, plant 2 along a diagonal.
- For 6″ spacing with extra room and an interesting pattern, plant 3 in a triangle.
- For an exact 6″ spacing, plant 4 in a square formation.
- For a 5″ spacing that’s just a little tight, plant 5 in the ‘X’ pattern.
- For 4½” spacing, plant 6 as shown above. Switch between double-^ and double-ν patterns for variety.
- For 4″ spacing with extra room, plant 7 or 8 as shown above.
- For exactly 4″ spacing, plant 9 in a square formation.
- For exactly 3″ spacing, plant 16 in a square 4X4 formation.

Here are two images to suggest how to make your own patterns inside a square foot. First, I made a one-square-foot sheet of paper. Then I cut circles from colored paper. In this case, I was working on the diagonal.

The red circles fit from corner to corner, but bulge out from the square. That won’t work for raised bed gardening, but might be alright if you plant directly into the ground.

Here are three green circles that do fit, although they waste a bit of the diagonal. In both cases, there is room for two more circles, representing five plants in the ‘X’ pattern.

Be more creative by fitting different sizes of circles into the square, for a mix of plants with varying spacing requirements.

## Applied Geometry for the Overall Garden

The usual geometry for a square-foot garden is a rectangle with a double row of 1-foot squares. That’s certainly the simplest wooden frame to design and build, especially for container gardening. The usual length ranges from 4 to 8 feet.

Remember the advantages of square foot gardening. First, planning and completing a single square foot of garden at one time is a small task. Second, it’s easy to reach across a two foot wide garden, and work without stepping on anything. That way you avoid compressing the soil.

Let’s apply some creative thinking to these principles. My own back yard has a curved driveway, with a low fence to avoid cutting the corner too sharply. So we planted a garden to follow that curve.

You could design a garden to fit into any corner of your lot, or to follow a path through the lot. You could make a spiral if you wish, leaving room for the footpath. The geometry can fit anywhere, along any curved or angular shape.

You could even opt for a series of circles rather than a rectangular grid of squares; or a pattern of triangles or diamonds. Just remember to make it easy to work on each bit of the garden from a footpath.

The next planning step is to map your garden on paper. Draw it to scale, unless you want to cover a floor with a life-size map made of butcher paper. A scale of one-to-twelve lets you map one foot of garden to one inch of ground. Small paper cut-outs allow you to envision changes quickly and easily.

If you don’t know how to make curved or odd-shaped wooden frames for container gardening, simply create prepare the ground in the shape you want. You can still apply the geometry to make the plants easy to reach; it’s just more difficult to prepare the soil as precisely as Bartholomew recommends.

## Applying Geometry in Your Design

So two ways to apply geometry to your Square Foot Garden are in your overall design, and in how to pack your plants into each square foot. Using these methods will help make your Square Foot Garden a creative success.

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