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STEP Prep Thread 2017

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Original post by Injective
unit sphere in R3R3 is equal to the set

{(x,y,z)∈R3∣x2+y2+z2=1}


{(x,y,z)∈R3∣x2+y2+z2=1}
the unit ball in R3R3 is equal to the set
{(x,y,z)∈R3∣x2+y2+z2≤1}.


Edit: my latex failed.


I know I've posted this elsewhere but it's so cool that I must let everyone know: a very cool result concerning unit n-balls is given at the bottom of this page: https://en.wikipedia.org/wiki/Gelfond%27s_constant
Original post by Injective
Two dimensions.


Calm, a circle is a special case of an ellipse right? My geometry is risty(non existent)


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Original post by drandy76
Calm, a circle is a special case of an ellipse right? My geometry is risty(non existent)


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Same man - my geometry is not so great. But I'm going to practice for the BMO, so it has to be good by then.
Original post by Injective
Same man - my geometry is not so great. But I'm going to practice for the BMO, so it has to be good by then.


My tutor this year is really good at geometry so hopefully she can rub off some of her talent on me


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Original post by IrrationalRoot
I know I've posted this elsewhere but it's so cool that I must let everyone know: a very cool result concerning unit n-balls is given at the bottom of this page: https://en.wikipedia.org/wiki/Gelfond%27s_constant


Ah cheers for that - I've just started preparing for STEP, so I'll see how much I can understand. :biggrin:
Edit: wow that's a nice result.
Original post by drandy76
Calm, a circle is a special case of an ellipse right? My geometry is risty(non existent)


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I think circles are those objects you get when the distance to one fixed point is a constant ratio to the distance to another...
Something like that...
Original post by drandy76
Calm, a circle is a special case of an ellipse right? My geometry is risty(non existent)


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Oh, yes to answer your question - it is a special case of an ellipse.
Original post by EricPiphany
I think circles are those objects you get when the distance to one fixed point is a constant ratio to the distance to another...
Something like that...


I think it would be a closed set of points equidistant from a fixed point? I'm sure I'm missing something though


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Original post by drandy76
I think it would be a closed set of points equidistant from a fixed point? I'm sure I'm missing something though


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There was this guy, a long time ago, called 'Apollonius'... probably a great guy, but I never met him.
https://en.wikipedia.org/wiki/Circles_of_Apollonius#Apollonius.27_definition_of_a_circle
Original post by drandy76
Calm, a circle is a special case of an ellipse right? My geometry is risty(non existent)


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And straight lines are a special case of circles, IIRC (which I probably haven't)
Original post by EricPiphany
There was this guy, a long time ago, called 'Apollonius'... probably a great guy, but I never met him.
https://en.wikipedia.org/wiki/Circles_of_Apollonius#Apollonius.27_definition_of_a_circle


He is my new hero


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Maybe you could use differential geometry to define a circle as a curve with constant curvature and zero torsion?
Original post by Mathemagicien
And straight lines are a special case of circles, IIRC (which I probably haven't)


Any clue where you might've read that? I'm curious now


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Original post by Alex:
Maybe you could use differential geometry to define a circle as a curve with constant curvature and zero torsion?


What would an object with constant curvature and non-zero torsion look like? Or would it largely vary?


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Original post by drandy76
What would an object with constant curvature and non-zero torsion look like? Or would it largely vary?


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A helix would be a good example.
Original post by Alex:
A helix would be a good example.


Would a helix be constant in both properties?


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Original post by drandy76
Any clue where you might've read that? I'm curious now


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Well, after doing some STEP problems about complex mappings (the mobius transformation?), it mapped circles and lines to either a circle or a line. And I thought to myself, why not consider lines as circles too? They have the same style of definition in the complex plane. That'd save me the effort of calling them different things. So I looked it up, and found some guys on the internet who agreed with me (https://en.wikipedia.org/wiki/Generalised_circle)
(edited 7 years ago)
Original post by drandy76
Would a helix be constant in both properties?


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Yes it would. You could make the torsion vary by making the helix an exponential helix. You could make the curvature vary by making the helix a conical helix.
Original post by Mathemagicien
Well, after doing some STEP problems about complex mappings (the mobius transformation?), it mapped circles and lines to either a circle or a line. And I thought to myself, why not consider lines as circles too? They have the same style of definition in the complex plane. That'd save me the effort of calling them different things. So I looked it up, and found some guys on the internet who agreed with me (https://en.wikipedia.org/wiki/Generalised_circle)


I'm lacking some background info to take this in properly, which means you've just given me something to do on my commute tomorrow


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Original post by drandy76
He is my new hero


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Original post by drandy76
Any clue where you might've read that? I'm curious now


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I you take Apollonius' definition seriously, a line is a circle (take the ratio in the definition to be one).
There's a applet here that you can use to visualise this.

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