Row Caterpillar

There are many season extension methods, but the simplest and easiest to implement by far, is the caterpillar half dome.  In order to determine the proper spacing of our soil beds before construction, we need to calculate the space requirement for this season extension method so that our beds have enough room and spacing to acomodate equipment involved in season extension.  This is where our Mathematical Formulas come into play.  Consider the diagram below.

Using the caterpillar half dome method for covering the soil bed, we know a couple of things.  We know the width of our bed is 4'.  We would also like to include some passive solar heating.  This can easily be done by adding some black 55 gal plastic drums that are filled with water.  The drums are setup outside the soil bed, to the north side.  This allows the drum to absorb the sun's rays during the day, heating the contained water.  At night, the water passively radiates the absorbed heat back into the atmosphere.  Since it is enclosed inside our caterpillar cover, it heats the air and soil inside thus keeping it warmer than the surrounding air which combats frost.

That said, these drums measure 35" tall and 24" in diameter.  That translates to roughly 3'x2'.  We know that we need to add the 2' diameter of the drum to the base diameter of the caterpillar, but how far does the caterpillar have to extend beyond the drum in order to clear it's 3' height?  Mathematics teaches us that the height of the drum, z, is equal to the square root of x times y.  When we solve for x, we get to 1.5'.  So now we know the width of our beds need to be 4'+2'+1.5' i.e. 7.5'.

Knowing that our diameter for the caterpillar is 7.5', we can solve the radius (d = 2r or r = d/2) which is to say the height of the caterpillar off the ground, at 3.75'.  To make things simple and provide some margin for adjustment, we round up the diameter to 8' and the radius or height to 4'.

In order to determine the size of clear plastic that we need to use to cover the caterpillar, we need to solve the circumference, or more accurately for the caterpillar, half the circumference.  Using another mathematical formula we calculate pi times diameter divided by 2 in order to come out with 25.13' which is the width our plastic will need to be to wrap over and cover the caterpillar.  We already know the length is the same as the bed, which is 12'.  That means that a 26'x12' sheet of clear plastic would cover the caterpillar, but what about the ends of the tunnel?  These have to be accounted for in determining the size of our plastic sheet to use.

The general rule of thumb is to angle the sides inward by 45 degrees until the ends all meet, and then tie it together and stake it.  But how much extra plastic is required to make this staking?  Again, we revert to mathematics, trigonometry to be exact.  Consider the diagram below.

In this diagram, the 8' side at the bottom, is the mouth of the caterpillar.  From our previous calculations, we determined it would be 8' wide.  Trigonometry teaches us that the corners of a triangle adds up to 180 degrees.  We already know that we are angling the plastic inward by 45 degrees so that would make the angle of the corner formed at the stake, 180 - 45 - 45 = 90 degrees.  Therefore, a straight line from the stake point to the mouth of the caterpillar, would divide the width of the caterpillar in two at the same time as dividing the stake corner in two and forming another right angle with the caterpillar mouth line.

OK, I know that can be confusing, but bear with me.  We need to know how much longer the plastic needs to be than the 12' length of our soil bed.  Therefore, we need to solve the value of Z in the diagram.  We also need to know how much further the beds stick out on the sides in order to plan the position of other beds.  For that we need to solve the value of X in the diagram.

We know the angle of C is 45 degrees and we know that B is a right angle (90 degrees) so that would make the angle of A equal to 180 - 45 - 90 = 45 degrees.  We know the length of Y is half of 8' so that makes it 4' long.  Trigonometry teaches that a right angle with equal opposite corners at 45 degrees has equal length legs thus we know that the length of X is equal to Y so X is also 4' long.

Now that we have the lengths of X and Y, we can calculate the length of Z using the trigonometric formula for calculating right triangle leg areas which states that:

Z² = X² + Y²

That makes the square of Z equal to 16 + 16 = 32 and when we square root that, we get 5.66' which is the length of Z.
Now we know that each mouth end is going to need 6' (rounded up for convenience) extra so doubling that for the two ends gives us 12' extra.  Incidentally, the shortest we could make this overshoot piece would be half the width of the caterpillar mouth which is 4' so 8' for both ends.  The reason we do it at an angle is to provide a gradient for breaking the wind.  If the ends of the caterpillar were just enclosed with minimal material, it would form a flat surface which could provide too little resistance to a wind gust, resulting in the caterpillar being scooped and blown away.

So now we know that we'll need a 26'x24' sheet of plastic to cover one 12'x4' soil bed WITH passive solar heating.