Thinking in different dimensions

[NewScientist]

I was wondering about something today,

I once heard a quote from Roger Penrose: a bar maid asked him what drink he wanted and he didn';t respond for a couple of seconds. When he did he said
'Sorry, I was thinking in about 12 different dimensino and I find it hard to come back to one' (or words to that effect)

What does he mean by different dimensions?

-NewScientist

 

[mathwonk]

well when you see a perspective drawing of a cube on a flat (2 dimensional) blackboard, what do you mean by thinking of it in three dimensions?

and if you understand how to find the volume of a 3- sphere by integrating the area formula for circular 2 dimensional slices of it, can you imagine how to compute the 4 dimensional volume of a 4 - sphere by integrating the volume formulas of 3 dimensional slices of the 4 sphere?

 

[Loren Booda]

Does zero dimensionality have any perspective?

 

[mathwonk]

i don't understand.

 

[vsage]

Who knows what he meant when he said that but thinking in more than three dimensions might be easier than you think for very simple objects. Take http://www.imagedump.com/index.cgi?pick=get&tp=265951 for example. Is it a 2d pentagram or is it the 4th dimension equivalent to a triangle? But you didn't ask about that.

It's possible he could have meant exactly what he said I guess. You might have trouble with what a dimension beyond 3 or 4 would look like though, so he must have been thinking pretty darn hard if he was up in the 12 range! Try thinking about some abstract concept for awhile then do a regular old task or try and carry on a regular conversation.

 

[mathwonk]

i don't understand this either. help! I am senile.

 

[Icebreaker]

NewScientist, you need not take this anecdote too seriously. What he meant was simply that for someone so used to abstraction, it's hard to come back to the concrete world of the ordinary senses.

 

[Loren Booda]

A three-dimensional space has perspective, and may be created from zero-dimensional singularities which by themselves have no perspective. Whence comes the relativity? From the (3 dimensional spatial) observer, of course! A sword by any other name.

 

[NewScientist]

Thank you all for your replies - I wasn't losing sleep over it :P!

And IceBreaker - you have a good point but I prefer to put the image of Penrose slightly more colourfully! Let us remember he was the one who introduced Hawking to topolgy and worked with him for many years - so he must have something damn smart going on in his head!

Anyway, I thought it was a nice little story. Another story I like is this one.

Hardy was visiting Srinivasa Ramanujun on his deathbed and commented:

" I came in taxicab number 1729, quite a dull number. I hoped this is not an unfavorable omen."

"'No,' Ramanujun replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'"

I just think that's ace! Anyway, thank you all for your thoughts.

-NewScientist

 

[SGT]

Of course it is very hard to visualize a geometrical system of more then three dimensions, but in Control Systems it is a very useful tool.
Think of a mobile in a 3 dimensional space. This mobile has a position, a velocity and an acceleration in each space coordinate, so we can represent it in a state space of 9 dimensions.

 

[Loren Booda]

As there may be an infinity of time derivatives (position, velocity, acceleration, change in acceleration, etc., ad infinitum) there may be an infinite number of such dimensions.

 

[SGT]

As there may be an infinity of time derivatives (position, velocity, acceleration, change in acceleration, etc., ad infinitum) there may be an infinite number of such dimensions.

You are right, but in general there is no practical reason to go beyond acceleration. At most the acceleration change is modeled as a random process.