The Geometry of Addition

The sum of a bunch of numbers can be viewed as a dot product between the numbers arranged in vector format and the vector (1,1...1) of the same dimension.

Given three numbers a,b,c then (a,b,c) • (1,1,1)  =  a.1+b.1+c.1 = a+b+c

There is a angle between the vectors (a,b,c) and (1,1,1) which is a measure of how varied a,b and c are. It is a measure of dispersion.  For vectors of the form (a,b,c)=w(1,1,1), where w is a constant, the angle to (1,1,1) is zero and there is zero dispersion. 

Other choices of vector (a,b,c) other that a=w, b=w, c=w there is a non-zero angle with the (1,1,1) vector and some dispersion.

That indicates some kind of filtering behavior, which should not be too surprising since summing is directly related to averaging and averaging most people can understand as some kind of filtering.

In physics the is the idea of conservation of energy. Which often is just balancing sums of squares.

For the vector (a,b,c) = (2,2,2) the energy is 2*2+2*2+2*2 = 12. 

The length of (1,1,1) is sqrt(3) with is longer than the length of of one of the basis vectors eg. (1,0,0).

For energy balance you have to account for that with a scaling factor of 1/sqrt(n). In the 3 dimensional case 1/sqrt(3).

Then the sum of the elements of (2,2,2) is 6.

Scaling gives 6/sqrt(3).

Squaring gives 36/3=12. Energy balance.

For inputs other than w(1,1,1) energy is lost (filtering action).

For example (1,2,3)  has energy 1+4+9=14

(1,2,3)• (1,1,1) =6

Scaling gives 6/sqrt(3).

Squaring gives 36/3 = 12

The original energy was 14, after filtering (summing) it is 12, so 2 units of energy were lost.

The lost energy could be recovered by projecting (1,2,3) onto vectors orthogonal to (at 90 degrees to)  (1,1,1).

This leads to a search for vectors orthogonal to (1,1....1).

For dimensions that are some positive integer power of 2 this naturally leads to orthogonal vectors like (1,1,-1,-1) and (1,-1,-1,1).

And these are just sum and difference equations eg. (a,b,c,d) • (1,-1,-1,1)=a.1+b.(-1)+c.(-1)+d.1=a-b-c+d.

Pinning the vector (1,1,...,1) is an indirect route to the (fast) Walsh Hadamard transform, which conserves energy after a scaling factor is applied. Conserving energy basically indicates that vector length is left unchanged. It might not be surprising since the Walsh Hadamard transform simply performs a change of basis, but a very odd and useful change of basis.