[sdiy] solid geometry

[Magnus Danielson]

From: "John L Marshall" <john.l.marshall at gte.net>
Subject: [sdiy] solid geometry
Date: Sat, 3 Jan 2004 12:40:56 -0800
Message-ID: <000901c3d239$e428e500$6a01a8c0 at cheetah>

> Math Gurus,

Dear John,

> I am building a spherical speaker to obtain the impulse response of acoustic
> spaces. I have a sphere with circumference of 38.25" and 25 speakers that I
> believe are suitable.

I would mount them in a single line (being somewhat "to blame" for the
spreading of line speakers in the PA-part of the world).

> The speakers are 3.5" diameter (will surface mount in a 3.0" hole)  with
> paper cone and rolled rubber surround. I measured the free air resonance of
> one speaker to be 110 Hz. These ancient ears could hear sound from the
> speaker up to 15 kHz.
> 
> I want to mount the speakers evenly distributed on the sphere. What is the
> formula for doing this?
> 
> I have found formulas for dodecahedrons and icosahedrons but they describe
> 12 and 20 flat surface solids.
> 
> Math wizards help me out.

I enjoyed the problem. A 25 speaker setup is not as easy to figure out as a
24 speaker setup. The 24 speaker setup is relatively trivial however...

Now, let's first break up 24 into suitable pieces... 24 = 2 * 2 * 2 * 3

Hmm... notice that we got 3 2' and that we are discussing the 2D surface of an
3D-object, namely a sphear. If we break up the sphere in 2 once for each
dimension we end up with a spearical surface similar to that of a point source
wavefront as it emits it from a floor-wall-wall corner of a room. Our mission
now is to locate 3 speakers in that sphere so that they cover an equal a amount
of space i.e. equal amount of steradians. By dividing this surface into three
equalently shaped and sized surfaces we acheive this. We can easilly do that
by picking a point being equally distanced from the axis, i.e. 45 degrees of
from any of them. From this point we can now draw lines up to the walls and
floor so that they is 45 degree from the floor and walls. We have now created
three 4-cornered shapes. In the middle of each such shape we stick a speaker.

Now, after this little exercise, what have we got? As we draw the full sphere
in our heads (which is easy if you got this far) we will have 4 rings of
speakers, 4 in the top and bottom rings and 8 in each of the two middle rings.
Naturally the speakers is evenly spread on these rings, so we have 90 degree
separation on the top and bottom ring and 45 degree separation on the middle
rings. Also, we have a 22,5 degree separation between the rings themselfs.

Now, as we turn the ball, we will see that these rings goes in three different
directions around the ball!

It should be possible to come up with the full coordinates from this
description, but let me know if you end up having trouble never the less.

BTW. No spheres, balls or speakers where hurt during the making of this email
or solution. Due to environmental aspects only reused thoughs have been used
in the preparation of this solution in order to minimize the amount of new
thought needed to solve the problem.

PS. I don't even know where I have a sphere suitable for study lying around.

Cheers,
Magnus