studiot

No it was a joke and quite funny I thought.

I have to observe that TIS is correct here. I meant to add exactly this to my post but it somehow got lost. The verb "to be happy" is a reflexive verb, so the subject and object are the same. So both are clear if someone says "I am happy". On the other hand, the verb to lie is transitive as it requires an object (the subject of the lie). So the question "about what ?" is valid, unless the subject has already been stated, as I said above. The paradox is a somewhat artificial construct in that people don't normally lie about everything.

There is also no paradox if you think harder about the physics of the situation since grains of millet are indifferent to sounds as they have no ears. pun intended

I think this is a tad unfair as TIS has the germ of the resolution, but has just not stated it clearly. However in view of the reception I got in the last thread about these ancient paradoxes, I am disinclined to help further.

+1 for understanding the point. and "I am lying" doesn't refer to anything. The conventional way to phrase this is to first classify everyone as either liars or truthers in everything they say. Then to offer the statement I am a liar.

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?