Hi,

I want to make a box with the relationship between the three dimensions based on the Golden Ratio, \phi. The problem is, I need to find the sizes for the external dimensions based on a set volume.

a = shortest side
b = a\phi
c= a\phi^2

If we say t is the material thickness I can calculate the volume of a known size box by the following formula

V = (a-2t)\times(b-2t)\times(c-2t)

What I am struggling to do is transpose that lot to give me the size of the box for a given volume. So far, I have replaced b and c and got this

V = (a-2t)\times(a\phi-2t)\times(a\phi^2-2t)

That structure looks like it should be familiar, but I can't think what I need to do to get this in the form of a= whatever. Any help would be appreciated.

As you know the Golden Ratio is 1.618.

The approximation I am going to suggest
for your purpose is a box of volume 1620
which is very close.
This would give you the following sizes.
9 * 12 * 15 = 1620
The shortest side being 9.

Well, what you have here is a 3rd order equation in 3 variables. Not a pretty thing. A mathemation might have fun with it, but an engineer's approach is how can I get the correct answer in a reasonable way. (Apologies to my old math professors, if any are still around).

Looking at your problem definition, you really don't have 3 variables. You are using the golden ratio as a variable, but it is a constant -- a known number and thus it's square is also a known number. That gets you to a 3rd order equation in 2 variables, closer but still no cigar. If you'd allow that in a given situation you would know the thickness of available materials, you could make t a constant - and solve individually for different values of t. That's what I would do (actually, I might just set up a table in excel, which would be quicker than solving the equation).

But, for the solution to 3 decimal points:
Golden ratio = 1.618; squared = 2.618
a = shortest side
b= 1.618 *a
c= 2.618*a
t= thickness of material

V=(a-2t) * (1.618a - 2t) * (2.618a - 2t)

multiplying (first, the first two)

V= (1.618 a^2 - 5.236 at + 4T^2) * (2.618a - 2t)

finally:

V= 4.236 a^3 - 16.944 a^2*t + 20.944 at^2 -8t^3

I left out the combining terms steps (1 in each multiplication) because I assume you know how to do that.

So, now you just pick a value for the thickness of your material, adjust the constants for that value and you have a solvable cubic equation. You can then plug whatever value you want for V and solve for a.The solution to the cubic equation will give the 3 possible mathematical values for a, but only one will make sense.

You then calculate b and c by applying the golden ratio and its square. To check your answer, subtract double the thickness from each and multiply the three results together to check that this equals your target volume.

A couple of examples:
Let t=1
then
V= 4.236 a^3 - 16.944a^2 + 20.944a -8

pick a value for V and solve for a:
For V=100
a=4.249 ----- minus 2, equals 2.249
b=4.249*1.618=6.875 -------minus 2, equals 4.875
c=4.249*2.618=11.124-------minus 2, equals 9.124

Actual V = 2.249*4.875*9.124 = 100.034

If that isn't accurate enough for you, use more decimal points. This should work for all values of V

Another example:
Let t=.5
then
V= 4.236a^3 - 8.472 a^2 +5.236 a -1
For V=1,000
a=6.852 ----- minus 1, equals 5.852
b=6.852*1.618=11.087-------minus 1, equals 10.087
c=6.852*2.618=17.939-------minus 1, equals 16.939

Actual V = 5.852*10.087*16.939 = 999.893

If the box was a cube, would 1 side be the root of the cube?

Well, his initial definition stated the ratio of the sides as 1:(golden ratio):(golden ratio squared). That is a rectangular solid not a cube.

If you had a different problem definition, which I think is what you are saying, of find the external dimension(s) of a cube made of a material with thickness t given a desired internal volume V, it is a much simpler problem.

The internal dimensions (a) would be equal to the cube root of V and the external dimension(s) would be a+2t.

If you actually want to build it, this doesn't give you the size to cut all of the pieces. You'd need 4 pieces cut to a and 2 pieces cut to a+2t - unless you want to get fancy and bevel the edges so all 6 pieces are the same.

Oops, that's 2 sides "a", 2 sides "a+2t", and 2 sides that are rectangles "a by "a by ".