So, I have been busy again. The F-Link card at the bottom was tricky, and is not a usual type of wow-card…
Usual hat tip to http://www.ianrowland.com – the man who got me going!
This is a little difficult to see, but is a representation of an impossible triangle that many mathematicians are familiar with.
Tricky, geometrical cut, but very rewarding to see made!
This was a nice, straightforward cut, if a little crap in the execution in this picture. Better ones will follow, but the concept is complete…
This is a nice cut, that is usually completed by the use of glue, but this is achieved without any glue at all! Tricky cut, but very rewarding to see that the cut works, and gives the correct and convincing ‘illusion’ 😉
Enjoy working this one out!
This was a fore-runner for the F-Link card, and uses the same solution in a slightly different way. Enjoy working it out – was quite tricky in places.
And now – the F-Link – Enjoy it! Took a while to work out precisely how to do it!
Well, that was quite a rewarding cut! Hope those were enjoyable! Have fun solving…
Being a musician and delving into mathematics, I am becoming more and more aware of the manner in which the way we do something, both literally and in the sense of mental approach, affects the cumulative results of any given activity.
We consider that we can do something, and so when we meet something that we are presently unable to, we say we can’t do it. “I can’t do vibrato…”, “I can’t see the solution to this maths problem…” These and infinite variations thereof, are commonplace assertions by students, it seems of every kind and level.
I have tried to cajole students into abandoning the word ‘can’t’ from their respective vocabularies. The result is initially a substitution; students find more inventive ways of telling themselves that they cannot do something. Then, the breakthrough; they get on with it, and stop faffing around with the quasi-poetic construction of phrases to the contraposition of ability, and instead accept that this thing they ‘can’t’ do they are going to try to do.
The next phase here begins, and we see Foucault’s “alembic of understanding” manifest itself; they crane their bodies and minds. Fidget. Squint. Reposition. The alchemy of technique has begun, and unless they are given the option to stop, the students persevere until the miraculous occurs; they CAN do it.
So, is success as simple as a change of approach?
Answer: No. But it bloody well helps.
Well, I have been setting my main business website, and have included a full gallery of the current set of wow cards!
Well, following on from going self-employed, I now have a website set up and running! It’s very basic, and uses the WordPress engine (it makes life a little *too* easy…) at the moment.
Comments and thoughts welcome!
I would like to thank you for your interest over the past couple of weeks. It’s been wonderful. I really would like to thank you for all your effort in terms of delivering the most exquisitely barbarous agony in my mouth. In so short a time, you went from being a functional part of my dental matrix of food-crushers, to an outpost of Hades itself.
Well, today you are being drilled and emptied in a process called ‘root canal’ (no idea what trees and waterways have to do with it, but ho hum…). At the end you will be dead. I would like you to know that this was totally personal, and kindly sod off…
Lots of love, M (-outh)
The simple set was an attempt prior to the Friedberg-Muchnik theorem to solve Post’s problem; Are there degrees that lie -strictly- between 0 and 0′?
One of Post’s later attempts was to construct a set called the Simple Set.
is Simple if it’s complement
is immune if for every program
is infinite (that is, the set of all values for which our program halts with an output is infinite),
So what do these definitions mean? Well, these definitions point to a set
that is simple and has some pretty nifty properties:
- They are enumerable – that is, they can be ‘enumerated’, i.e. we can discover which numbers are members, by means of a Turing Program (or, rather, a computer)
- The complement of (that is, the set of all things not in ) is infinite.
- The simple set ‘touches’ every infinite enumerable set. That means, that for any infinite set that is enumerable, our simple set shares at least one number.
What is interesting is how we create this set. We start by ‘numbering’ all of our enumerable
sets, so that
, that means that every enumerable set is different.
We now go through each set with a goal that changes for each set, but remains more or less the same. Post’s original idea was to ask ‘does this set have a member that is greater than twice it’s
number?’. This means that the number we choose from
will be different from
, etc. Once we have a number from some
then we move on to
and so on. So, for any
it either won’t give us some number from within it (as it won’t have any numbers in it that are larger than twice it’s
number) or will give us at least one number.
So, you can see that the procedure is quite straightforward, and yields this set with some rather interesting properties.
EDIT: I should have stated, the simple set didn’t
solve Post’s problem! It was shown, for those who know, it was shown by Martin (1966) that all effectively simple sets are complete; if
is simple, then