Chris S. writes:

I was wondering what is the theoretical ‘area’ of contact between two spheres in contact with each other. I was unfortunately not able to locate much (if any) information on this. After some thought into this I’ve realised that the spheres would meet at a single ‘point’ however what would the area of this ‘point’ be? The only source related to this claimed the area of contact, the point, has no area. How can a point have no area? If the spheres touch, musn’t there be an area shared between them? Even if only one atom?

Hi, the issue here is that there is a vast difference between physical, real things and the mathematical ideas that model them.

Real, mathematical spheres don’t exist, plain and simple! Never could, even as a region of space— space itself has a granularity (apparently) at a scale of about 10^-33 meters. There simply cannot exist a perfectly spherical region in physical space, much less a perfectly spherical body.

But as an abstraction, the idea of a sphere is very useful: lots of things, quite evidently, are spherical for all practical purposes.

For that matter, “points” don’t exist either, and are also a mathematical abstraction. (So, too, is “area”. Real things are rough, bumpy and not at all like continuous surfaces, on a fine enough scale) But again, these _ideas_ are very good at getting at something important about lots and lots of physical things, and so have proved useful.

Tangent spheres do indeed meet in a single point, which has no area.

Spherical things meet in some other, messier way.

Hope this helps!

After taping this segment, Kyle and I discovered an incredible way to make LOTS of money. We will extend this amazing opportunity to our listeners next week!

Standard Podcast [ 7:41 ] Play Now | Play in Popup | Download

Given a difference table, as we considered back in EV. What’s the Difference , how do we come up with a polynomial that gives the values on the top row?

For example, suppose we have

-1     -1     3     35     143     399     899 . . . . .
      0     4     32    108     256     500  . . . . .
         4    28    76     148      244  . . . . .
             24    48     72       96   . . . . .
                  24     24    24    . . . . .

What is the polynomial P(n), of degree four, that gives

P(0) = -1 P(1) = -1 P(2) = 3 P(3) = 35 P(4) = 143 , etc.

Can this be expressed simply in terms of the leading values on the left of the table: -1, 0, 4, 24, 24?

Read the rest of this entry »

For me the holidays end on Monday so I just have time to post this seasonal question.

Why is Christmas the same as Halloween?

Specifically why is Oct. 31 = Dec. 25.

Hint: you need to look at exactly how this is written.

A short problem with a long pedigree.

I found this in Martin Gardner’s book ‘The Colossal Book of Short Puzzles and Problems’, problem 3.9 in my copy.  He credits Solomon Wolf Golomb, the inventor of Polyominoes which inspired tetris.

Isaac Asimov based a whole story on this puzzle,  ‘A Curious Case of Income Tax Fraud’, part of his Black Widowers series.

I asked some work colleagues and had some amusing answers, none of which were maths related.  Maybe you have your own?

Enjoy!

Steve

Happy New Year Math Factor fans.

My name is Stephen Morris and like Edmund I’m going to be posting in the Math Factor website.  I have been a listener for some time and have posted comments as ‘stevestyle’.  I’m not a professional mathematician, just a keen amateur and fan of math puzzles.  I live in England.

I thought I would start with an anecdote from the recent holidays, where playing scrabble with the family helped with one of Chaim’s problems.  I hope you enjoy it.

Read the rest of this entry »