with Alicia Boole
Starting at eighteen, with little specially coloured cubes, that had been devised by a regular visitor to her home, Charles Howard Hinton, to visualise the four dimensional Teseract which he had named, Alicia Boole was able to develop a phenomenal ability to visualise four dimensions. She went on, as Alicia Boole Stott, to publish papers on the subject, share her models and collaborate with other mathematicians.
In the same way that we cut through three dimensional shapes with an infinitely thin knife in order to visualize their two dimensional cross-sections, and perhaps with a lot of practice, and the right use of angles, we can work backwards from cross-sections to reproduce the solid, Alicia was able to mentally cut through four dimensional polytopes to see the three dimensional shapes that are their sections. She was able to see these polyhedra change size and for new three dimensional shapes to appear and disappear as she cut further through the four dimensional object in her mind. Alicia then drew what she saw and made nets and models to explain the 4 dimensional polytopes she visualised to other people.
Given the fact that her father was the logician Boole, of Boolean logic fame, and her mother was related to the Everests of Mount Everest, it seems likely that Alicia could have accomplished more if her father had not died when she was four years old and she was plunged into penury, and it may have helped if the education system had provided better formal education to the females of the time.
However, she certainly received a good tuition from her mother. Mary Everest Boole had studied with her husband, George Boole. When Boole died, Everest Boole moved to England and was oﬀered a job at Queen’s College in London as a librarian. Her passion however was teaching, and she liked giving advice to the students [Mich]. She had innovating ideas about education, believing for example that children should manipulate things in order to make the unconscious understanding of mathematical ideas grow [Mich]. Her belief that models should be used in order to visualize and understand geometrical objects is reﬂected in the following words:
There is another set of models, the use of which is to provide people who have left school with a means of learning the relation between three dimensions and four. [Eve1] The geometric education may begin as soon as the child’s hands can grasp objects. Let him have, among his toys, the ﬁve regular solids and a cut cone. [Eve2]
The Princess of Polytopia: Alicia Boole Stott and the 120-cell: http://goo.gl/cul3bd
Dissertation (open) on Alicia and her polytopes (from Groningen): http://goo.gl/igTCqK
And as a (closed) paper by Irene Polo-Blanco: http://goo.gl/xYQPaX
Using Cross sections: http://goo.gl/JqNE8l
Which is part of:
A fascinating guide to visualising 4D polytopes: http://goo.gl/Npmld5
Charles Howard Hinton: http://goo.gl/yS0SY2
So why can’t I put 4 straws together so that each one is perpendicular to the other 3?
Wow, this is amazing. I didn’t know that this was possible. Great post!
Thanks Science on Google+!
Jack Martinelli – are you serious, or are you being coy? If you’ve tried it, you certainly know the answer.
We’re lucky to have computers that can help us to visualize higher dimensions. I think it’s a lot more commonplace for people to be able visualize four or more dimensions today.
I think Alicia Boole Stott would have really appreciated that.
Bruce Elliott yup, observationally there are only 3 orthogonal dimensions. It can be misleading to teach a 4th orthogonal dimension for extra embedded dimensions. See
Dark Energy Solved to understand what I mean.
Wow I’ve never visualized for dimensions like this!
It is something that is not easy for more than 4 dimensions. In string theory we deal with a minumum of 6 microscopic dimens tightly curled up in Kähler Manifolds (of which Calabi-Yau is very beautiful and you can google it and look at youtube for a visualisation) so this is tricky. Then there are normal 3 spatial and 1 time dimensions to take care of which is okay.
But it’s getting there. String theory have advanced sufficiently to start making predicions that are testable in the standard model! We live in exciting times, albeit many wish the universe wouldn’t be fully this complex 🙂
Sorry Jack Martinelli – that (your YouTube) really makes very little sense to me.
I don’t know what you mean by “misleading to teach a 4th orthogonal dimension”. What’s misleading? It’s perfectly self-consistent mathematics.
Bruce Elliott Yes I agree. And can be useful. But does not represent anything you can see. At least in the same way you can see 3d cube.
Jack Martinelli – Well, that’s certainly true – I don’t expect to see (fully) any 4-D objects any time too soon!
Stott’s name has been revived in the Polygloss and the various sources that rely on it. The most recent is the project to find the johnson polychora (4d polytopes), which rely on my notation, and Stott operators.
wendy krieger – That sounds very interesting. Do have any links you can share?
Bruce Elliott The Polygloss is at http://www.os2fan2.com/glossn/index.html ,
I even get mentions on the page in the OP at the page on
http://eusebeia.dyndns.org/4d/octagonny , since octagonny is my name for that polytope.
His renders are quite good, i must say, but i can’t get the stereo effect to work. Still, i’ve been feeding him input into his project.
On the other hand, i don’t really do graphics. I draw rude diagrams from time to time, and let someone else pretty them up, or comment on other people’s diagrams to similar effect.
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