studiot

I'm glad you are thinking about the maths because you asked about surfaces and one of the difference between a sphere and a donut (or teacup or torus) is that you can shrink any circle (or any other curve that forms a single closed loop) to a point on the surface of the sphere but you can't on a torus. This article should be accessible. http://www.maths.ed.ac.uk/~aar/surgery/zeeman.pdfhttp://www.maths.ed.ac.uk/~aar/surgery/zeeman.pdf

That you have had too much Christmas spirit already? Seriously I was unable to determine your beef from that ramblng post. As regards paradoxes, I have not encountered all of them. The ones I have encountered are usually formed by combining two (or more) statements inappropriately. Resolution is achiveved by untangling them and taking one statemen at a time.

Have you tried some algebra? I make it [math]4\int_1^2 {\sqrt {1 + \frac{{\left( {{x^2} - 4} \right)\left( {{x^2} + 4} \right)}}{x}} dx} [/math]

Perhaps you should read this book (it takes between one and two hours) What is Random? Edward Beltrami

Perhaps you should read Conway's new thread to find the answer to that particular question.

Good to hear that. So what happens if you shrink your black circle to a point so that each zero radius circle describes a single position on the sphere?

+1 Yes most likely sulphuric (car battery acid). However hydrochloric is put into "limescale removers" for toilets, by bleach manufacturers. It comes in black rather than blue bottles in the UK, but is easy to mistake. The acid and tha alkali remove different componeents of a blockage. As already described, caustic alkali (yes it is caustic so use rubber gloves) attacks the grease componenent which sticks things together. But cellulose fibres in paper and cloth and other fibres such as hair and roots can be mechanically trapped in the drain. That is where you should use the acid to dissolve them out, as the alkali will not touch that situation. Always remove as much as possible mechanically first, since that will reduce the need for chemical action and is cheaper. If you do use the acid, a bucket of washing soda (sodium carbonate) will flush it away safely afterwards. The build up your referred to creates a highly nutrient soup for plants so they try to break into the drain with their roots. This can be quite a nuisance with old drains. Finally you should keep monitoring the situation and clearing before it gets to bad in future.

Are you not aware of this formula? [math]L = \int_a^b {\sqrt {\left( {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \right)} } dx[/math]

Well if this was a serious question and you want serious discussion how about also responding to my serious Physics comment? After all you have posted in quantum physics.

Yeah I agree what's the point ? Edit And what about halucinations, mirages , tromp l'oeil and other deceptions? Love the picture by the way. +1

Both have specific status amd meanings in Science, which are different from general usage. I hypothesis is a (hopefully) well thought out 'guess' or proposal. That is is is made for good scientific reasons. Once made the hypothesis is tested, preferably in many ways. If the predictions of the hypothesis conform to observation then it may be elevated to the status of Theory. If not it is rejected. Usually a Theory actually contains more than one hypothesis along with some development of their interactions and implications. Newcomers here sometimes fail to make this distinction and are marked down as a result. Does this help?

And use wolfram alpha to provide some known example results to test your program.

Do you know this website? Wolfram http://mathworld.wolfram.com/AckermannFunction.html By the way (BTW) since this is coursework we can't do it for you but we can hint and guide. However you must play your part and show some working. Also this should be in homework help.

The analysis commits the classic error of division by zero. Both the cosine and the cotangent are zero for angles equal to [math]\left( {2n - 1} \right)\pi \quad :n = 1,2,3...[/math] leading to division by zero in both the defining equations.

Some very good kitchen table descriptions here to take note of. I particularly liked the two above +1

Have you seen this? http://demonstrations.wolfram.com/ParametricEquationOfACircleIn3D/

Thank you for this, I will print it out to look at tomorrow. On first sight it appear more rigorous than this comment in your previous diagram/post There are no tangent planes in your diagram. A sphere has tangent planes, a circle has tangent lines.

Even with a much simpler shorter post you didn't read it properly. Like you my post mentioned four angles. Further it didn't say you didn't read my post it said you didn't read it properly. I didn't say it was. So what? I also asked a question about (the mathematics of) your diagram but instead of answering you accuse me of not reading your post. Considering I am not accusing you of being a crank but actually seeking to find out if you genuinely have a useful viewpoint to offer and trying to help you with it, despite your mathematical failings, this just extreme arrogance and disregard for others.

Keep probing. +1

Well I tried, even though I'm a bit sorry I bothered since you clearly didn't read my post properly. What do you think the black loop curve is in your diagram? Do you think that it could be used for spherical trigonometry?

Thank you for posting the asymptotic curve. Yes I often make this point that SR is undefined or unknown above c. Though I do not go as far as refuting SR. +1 There are other more prosaic phenomena in Nature and mathematics which show this behaviour (such as the specific energy line in fluids). These have more than one type of recovery once the critical value is passed.

I wonder if this is a key to your difficulty. I have hastily dashed of some sketches as I had to sneak onto the scanner. Sorry for the lack of quality. Each of the three shows an angle as described between two straight lines in different orientations relative to ayz coordinate axes. Each sketch shows what happens as you move the pair of lines and their included angle relative to each axis, but wihtout changing the angle itself or the relationship between the pair of lines. I wondered if this had any bearing on what you are trying to describe.

rIsn't this veering off topic? A surface is a two dimensional object, insofar as it may or may not be embedded in a space of higher dimension. You have referred to projective geometry. One of the aspects of PG is that it is indifferent to distance. That is why a diagram of Desargues Theorem works. http://mathworld.wolfram.com/DesarguesTheorem.html

So far no one has ridiculed you, though you have been asked for more detail. A short answer is that yes in some circumstance there are (serious) applications of probability to the physics of time. For instance given the probability of the radioactive decay of an atom in a bunch of atoms, you can calculate the probability of simultaneous decay of 2,3 or more atoms. You can also discuss the meaning of simultaneous in terms of the time taken and the uncertainty principle. For a longer answer you need to propose a less flippant example.

Why is it so important to you that your expression be an identity? It may well be an expression that correctly models some phenomenon. But an identity relationship does not depend upon the use you put the expression to. Either the expression always equal one, whatever you use it for and it is then an identity. Or the expression doesn't always equal one, in which case it is not an identity. I am not sure what you are trying to achieve with this projective stuff. Can you elaborate?