A five minute presentation on imagination in mathematics. I explore the idea of dimensions first of all through the 5th dimension via constructions of cubes, and I then ote that in projecting these we are encountering negative numbers. I then look at the coastline of Ireland as an example of an object with fractal dimension and finish off with the idea of fractal narratives. There are a number of embedded links in the presentation to relevant resources on the web, and slides have notes attached.
29. i cs !
ma t
at h e
M
here
ery w
Its ev
look
you
Thanks for your time,
Laurence Cuffe
Notas do Editor
The link between the colons leads to a musical rendition of the first 30 digits of pi.
This is a poem in mathematics
Maths is a language
As it’s a language, it’s a platform for verbal thinking. The bigger the verbal space we have at our command, the more we can think of.
I now Invite you to come with me and try and visualize the 5th dimension
A line, 1d, two zero dimensional points, linked.
Two lines, linked, 2D, a square
Two squares, linked, A cube, 3D
Two Cubes, linked, a teseract, 4D
I could join two of these, but the result would be hard to see. However it’s a case of ICT to the rescue.
When we want to apreciate the shape of something 3 dimensional, we take it up and turn it around in our hands.
We can do the same thing in 5 Dimensions. If you click on the word rotate above, it links out to an image of a penteract, a 5D cube rotating in 5 dimensions. Actually, it’s a 2D image, rotating so I guess time is a third dimension, and we can consider the two missing dimensions removed by the projection as negative, so now you are looking at two negative dimensions.
Moving on. A line is the shortest distance between two points.
I want to get from B&H photographic to the apple store.
The shortest distance,
Turns out to be a grid of lines. In this geometry, called Taxicab or Manhatten geometry, lines can overlap without crossing, and have area. Before we leave this I’m going to consider one more idea, a Circle
In this geometry, a Circle becomes a diamond shaped pattern of discrete points.
Ok we have had dimensions 0 through 5 as whole numbers, and we have also seen negate 2 dimensions. But we know more numbers than just whole numbers, so its now time for decimals. 1.22 is the fractal dimension of the Irish coast. A bunch of students in TCD found this out in 2010.
The dot links to a google image, but we will skip that. Lets look at some pictures
This is the coast and its wiggly.
Even when we zoom in, its still just about as wiggly,
And again
And again,
And again. As we zoom in we just keep on finding more detail. The coast of Ireland can be envisaged as being infinitely long with more detail apearing as we zoom in. Fractals are every where, tree bark, coulliflour, lots of places. But I want to give you another example, which is not geometrical
The link here is to the site fractal narratives, which provides stories which give you more detail each time you click on the space between sentences where it says do you want to know more. Most subjects, even life itself are like that, a fractal narrative with potentially an endless level of detail
These were the sites used, thanks for the time and If I can leave you with one thought,
