Distance from density - how far apart are my plants?
July 4, 2020 3:49 PM   Subscribe

I thought this would be easy - how wrong my tiny math mind was. I have a spreadsheet I've used smallpdf.com , imgur if that fails. for years to calculate how many plants to plant in an area based on some simple calcs. But for planting distance I've always just guestimated, so I've tried to work out how to derive planting distance from my table and after scribbling a lot have got ... nowhere.

Subset of my table, A are constants, B = 1/(A*A), C is the area I enter, D - B * C

A B C D E
Distance Plants per m² Area NumberOfPlants Plant distance
1.00 1.00 100 100 ?
1.50 0.44 100 44
1.80 0.31 100 31
2.00 0.25 100 25
2.50 0.16 100 16
5.00 0.04 100 4

What I want to do is have numbers of plants per m² as my constant and to output plant distance.
So is this simple? Is there more than one way to do it?
posted by unearthed to Science & Nature (9 answers total)
 
If I understand your question, you just need to calculate the square root of the area available to a single plant.

If x is the number of plants per square meter then the excel formula for distance in meters would be =sqrt(1/X).
posted by jon1270 at 4:11 PM on July 4, 2020


Simple example. Suppose you have a square garden of area 100 m², side length of 10m. Think of it marked off in 10x10 squares. You want to plant 100 plants in a square arrangement, so you put each one in one of the squares. Planting distance = 1m.

To generalize, the number plants is a row will be sqrt(number of plants), and the length of a row will be sqrt(area), so planting distance is sqrt(area) divided by sqrt(number of plants).
posted by SemiSalt at 4:11 PM on July 4, 2020


Working out jon1270's equation in more steps --

X plants per square meter, m^2, means that each plant gets 1/X m^2 to itself. If there are three plants per m^2, each of them needs a third of a square meter -- how big is that?

Here we run into the problem that if I say "one-third square meter" it sounds a lot like "one-third meter, squared" but they aren't the same. It's the area, square-meter, m^2, that we're dividing into thirds, not the sides, which are measured in meters.

The side of a square that's 1/X square-meters in area is the square root of 1/X meters long -- that's 1/(sqrt(X)) meters in length. In algebra,
   1            1         1  
------- m *  ------- m =  - m^2
sqrt(X)      sqrt(X)      X
This is only easy to draw for plants that fit in even numbers into a square meter, but if you try it with 1 plant, 4 plants, 9 plants per square meter, the centers of each plants' patch will be 1m, 1/2 m, 1/3 m apart.

Oh, and also;
               1
sqrt(1/X) = -------
            sqrt(X)

posted by clew at 7:54 PM on July 4, 2020


Thanks heaps jon1270, and SemiSalt, that gets me far enough for what I need.

clew, yes, I realise now this is only perfect for integer squares but it is fine for my purposes at the moment, it's now an intelligent rule of thumb!

I really just want to give a spacing distance guide to clients/planters and make it so simple for them that they don't need to cart an increasingly fragile paper plan with them.

I just mark (colour coded) the area, drop a coloured distance stick in the place and tell them to plant x number of plants there - and I can go and do other things. Planters really like it when they don't have to use plans.
posted by unearthed at 9:32 PM on July 4, 2020


It’s perfect for any density of plants, it just isn’t easy to calculate! Neat idea, making a custom spacing stick.
posted by clew at 12:51 AM on July 5, 2020


Using sqrt(per-plant area) for the length of your spacing stick will work if you're planting on a square grid.

But assuming that the area occupied by an isolated plant will be roughly circular with the planting location at the centre, then for the most even coverage you'd want to tile your planted area with hexagons rather than squares. To do that puts the planting locations at the corners of a tiling of equilateral triangles rather than a tiling of squares; each plant will have six nearest neighbours rather than four.

The relationships between the overall area to be planted, the planting density and the number of plants required all remain the same regardless of planting grid pattern, but the distances between individual plants are different for square and triangular planting grids. Side lengths on the triangular planting grid will be twice the hexagonal area tile's apothem.

The general formula for the area of a regular polygon is half the circumference times the apothem, and the apothem of a regular hexagon is sqrt(3)/2 times the side length i.e. sqrt(3)/12 times the circumference; so the circumference is 12/sqrt(3) times the apothem, half the circumference is 6/sqrt(3) times the apothem, so the area is 6/sqrt(3) times the square of the apothem, so the apothem is sqrt(sqrt(3)/6 times the area) and the planting distance is twice that, or 1.07457*sqrt(per-plant area).

So for a hexagonal planting pattern, you'd want a spacing stick about 7.5% longer than what you'd use for a square planting pattern with the same planting density.
posted by flabdablet at 7:44 AM on July 5, 2020


And it's just occurred to me that on any job site where you did want a hexagonal tiling, you could make both the intended pattern and the necessary spacing no-brainer obvious by using spacing triangles made from three spacing sticks bonded together.
posted by flabdablet at 8:01 AM on July 5, 2020


Interesting, flabdablet, currently in Q at dentist so my thinking is limited! but, that apothem/hex method looks like it might help account a bit more for edge effect, maybe

My schemes always involve irregular spaces, generally longer in one axis than the other. For many calcs I simply sqrt the area. Increasingly I'm moving to voronoi shape spaces ( for various reasons), and my planting advice is oriented to resulting in a random arrangement, which often assumes a voronoi-like 'pattern' - I ask planters to be approximate in their plant spacing. A few landscape people Earth-wide are trying to facilitate easy 'random' plantings without overburdening planters with rules.
posted by unearthed at 4:42 PM on July 5, 2020


In that case, your sqrt(per-plant area) distancing stick is probably all you need. If planters are told to try to avoid regular patterns but keep plants spaced between a stick and a stick and a half apart, then the density will probably end up near enough to right.
posted by flabdablet at 2:02 AM on July 7, 2020


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