The New Book
Part 1: Chapter 1
Hilbert Space

Visualizing multi-dimensional space

Human brains have evolved to cope with only a three dimensional world, so it is not possible for us to visualize a multidimensional space full of objects. We approximate some kind of picture by imagining the objects in a three dimensional space and seeing those objects as lists of features (parameter lists).

The problem with this visualization is that it doesn't provide any perception of scale. It isn't easy to see that every new dimension effectively multiplies the number of possible objects several times over. To do this, we have to use another mental model - one that can be provided by thinking about a credit card

There are twelve numbers on a credit card. These can be thought of as twelve dimensions in a twelve dimensional Hilbert space - where each dimension (number position on the card) can have any of ten different values.

Note: When a dimension or a parameter can have several different values, these values are called scalars

Say the ten possible numbers in the first position on the card (numbers 0 to 9) represent ten different colors. And the ten possible numbers in the second position represent ten different shapes. These first two positions will then be able to describe one hundred different objects, each being a different color and shape combination (i. e., if these two number positions on the card are thought of as describing two dimensions in a Hilbert space, the two dimensional space will contain 100 objects - each with a different combination of color and shape).

Now make the ten numbers in the third position represent ten different materials. You will then have a three dimensional space that holds a thousand objects each with a different color-shape-material combination.

If these objects could be owned by any of ten different people, the number in the fourth position could represent each of the ten people. This would give you a four dimensional space that describes all the possible ways each of ten people could own a different one of these objects - a space that would contain ten thousand locations, each location representing one the possible combinations.

The other number positions on the card could represent other information that may be relevant to the objects: which of ten countries they could be in; ten different kinds of boxes they could be contained in; etc. When all these twelve positions are allocated to features relating to the description of the objects, it would be describing a twelve dimensional Hilbert space that contained over one trillion different possible combination of the parameters.

It is useful here to look at this situation from the mind set of a computer programmer. To the programmer, this would seem to be an elementary problem to solve because it is very easy to set up a database that has twelve different fields and for every object you enter one of the ten variables into each field. Such a procedure is common practice in information technology.

However, database technology is based upon the fact that you know what objects you are dealing with beforehand, therefore you only have to enter the details of each of these known objects. As the known objects are usually a very small proportion of the total of all possible combinations, a computer memory can usually handle this quite easily.

If the computer programmer had to allow for a memory to hold objects with all possible combinations, it would require something of the order of one billion records (actually, twelve times ten to the power of ten = 1,200,000,000).

( Note: it is the magnitude of the total search space of a twelve digit credit card number that provides security against fraudsters who try to guess credit card numbers.)

Now, try to imagine a Hilbert space that has tens of thousands of dimensions, each of which can have hundreds of different values. The number of possible combinations is just unimaginable. No computer on earth could cope with this variety of combinations. Yet, this is the space that Nature has to search, to find the exact combination of genes that give birth to a human.

It seems an impossible task, but Nature has found an efficient way to do it.