3D graphing of mathematical functions

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Frances,

Thank for replying; but I think I answered your question in my opening post, and in my response to John ^^^.

• l did not do 'A level' Maths [my Grammar School could not understand that I might want to do Maths if I was in the Biology stream]
• I did not do an 'Engineering' degree ... I did Ergonomics with Human Biology; and my only advance on 'O level' Maths was to do some extra Statistics classes

I was hoping that in these enlightened times, there would be an interactive tutorial somewhere on the web, that started at a level which I could grasp without reading all those TextBooks and which would enable me to learn at my own pace; to the level that I need to reach to pursue my interest in the Radiolaria.

Apparently not.

MichaelG.

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https://en.m.wikipedia.org/wiki/Radiolaria

Edited By Michael Gilligan on 14/12/2016 22:06:29

Michael, I can't do it now and I'm busy all tomorrow but I'll try and post an explanation of what I meant by "expect by looking at the function". Given you have this tool now, you can probably learn a lot by playing with it for various simple functions and slowly build up your knowledge. I'd guess that Open University maths texts should also be helpful as they can't assume A level as a starting point for example?

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That would be great, John ... Thank You !

Given a desire to learn; I think maybe I just need to get my toe in the door.

MichaelG.

Michael, try this:

**LINK**

Thanks again, John

MichaelG.

Michael,

OK, so your equation is this (slightly rewitten).

The first thing is that the numbers, 0.75 / 5 / 5, don't really add anything. The 0.75 just scales the height of the fence to that value, and the 5's scale the values of x and y. A very common thing to do in maths in this sort of situation is to "normalise" the equation by removing scale factors if they can be restored easily if you want them but otherwise just clutter up the working. In this case we would just re-write the equation like this.

So first I've scrubbed out the 0.75 as we can just multiply whatever value we get from the equation later by simple multiplication. Second I've removed the 5s (or actually the 25s as the 5s are squared) - if you wanted the result for particular values in the original formula you'd just multiply those values by 25 and insert the result in the second formula. It already looks simpler. Next it's written as "e to the minus()" as 1/anything is anything to the power -1. That isn't essential but looks tidier.

You probably know that e is the so-called Euler's constant, or the base of natural logarithms. It's a bit like pi, turns up all over the place in maths. It equals 2.7182818284590452353602874713527....and goes on for ever.

e to the power of something, or exp(something), gets very large very quickly as the something increases, it's exponential. So exp(1) = 2.717...; exp(2) = 7.389; exp(5) = 148.4...; exp(10) = 22,000.5; exp(25) = 72,004,899,337.4! As we have the reciprocal of the exponential in our formula, it gets very small very quickly. But also note that exp(0) = 1.

So if x and / or y in our formula above were zero, the function is equal to one. x and / or y are zero along the top edges of the fence, so that's why the top is flat.

Now imagine an ant exploring the surface carrying a tiny GPS that tells him his coordinates including height. Suppose that the y axis points North and the x axis points east. (And the earth is flat!) Now the ant crawls eastwards along a line of latitude "1" from say x=-5 to x=5. Along that line, since y equals 1, we could write the equation:

At y=-5 the value of z is 1/72,004,899,337.4 = next to nothing! Since x is squared, z has the same value at x=+5. At x=+2 or -2, z is 0.018, so still pretty small. x=+/-1 gives z=0.37. You can see why we get the "fence" since the function gets very small very quickly.

If the ant crawled along y=2, the function would be

Since the value of the exponent is now 4 times larger, z is much smaller except when x is (if you like) 4 times closer to zero. This why the fence gets sharper as you get further from the origin.

Incidentally, the function in the third equation is, except from some normalising factors, the same as the statistical "normal distribution" or "gaussian function". So the original function is a sort of two-dimensional gaussian (but in the real 2-d gaussian x squared and y squared are added not multiplied).

I hope this explanation helps. Generally with most of the formulas you could apply similar technicques, by first getting rid of multiplying coefficients and then looking at the behaviour of the function if you keep one variable constant and vary the other, and do that for various "constant" values. In effect you will be producing cross-sections of the shape. The App should easily allow you to do that.

John,

Really good explanation but for us dumbos at the back would you please show step by step how you 'just' re-wrote the equation to get rid of the reciprocal?

I know you got it right but I wasn't paying attention during that algebra lesson, and a good few others!

Thanks,

Dave

Is the colour a representation of the z value or have you/can you add a simple 4th dimension with this program? (ie. colour - I can't think of many other ways to add a simple 4th dimension)

John,

Very much appreciated ... I will work carefully through that [to make sure that I comprehend], before asking for more.

I think [hope] that I can get to grips with things "this way round" ... But I foresee big problems coming if I have a particular shape in mind, and wish to write an equation that produces it.

The App is amazing, and with your encouragement I'm confident that I will at least improve my understanding.

MichaelG.

Dave, I'm sorry I can't, offhand! It's just convention that one can write exp(-x) = 1/exp(x). If you plug some numbers into a calculator you'll find it always works. It can probably be proved but I have a suspicion it needs complex variables and logarithms to do so (or at least logs).

That is certainly true! If you really have to do that it's an "approximation" problem and usually it will only be an approximation. Sometimes if you want a shape to do something (a cam for example) it might be easier to see if you can use a shape you can mathematically generate rather than find an approximation.

No problem John, I'll do some digging. I use conventions too, which is OK until I misremember one, or never learned it. If I find out I'll share the answer.

Might do it now, I've spent the afternoon downloading software only to find out it's the wrong package. A small sherry and a little maths might cheer me up.

Dave

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John,

It's more a matter of "wanting to" than "having to"

... as I mentioned earlier; the underlying motive for this exploration is that I want some understanding [or at least appreciation] of how biological forms develop,

A couple of years ago, I studied an original print of Mrs Bury's book 'Figures of remarkable forms' **LINK** https://blogs.princeton.edu/graphicarts/2007/09/one_of_the_great_botanicals.html

and I have been hooked ever since.

Alan Turing's paper on morphogenesis would [probably] explain a lot, if only I could cope with the maths ... but Turing didn't have the luxury of an iPad and the QuickGraph App, so the diagrams are rather sparse.

MichaelG.

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Edit: for info. here is Turing's paper ... http://dna.caltech.edu/courses/cs191/paperscs191/turing.pdf

Edited By Michael Gilligan on 16/12/2016 16:20:14

Got it.

where you lake the logarithm to base a. But the log of 1 to any base is zero, and the log to base a of a is 1.

Dave, it's just one of those things that flash past in a maths class - if you miss it, you're stricken forever-after. You can swap terms between numerator and denominator (or visa-versa) so long as you change the sign of the exponent.

Another is the counter-intuitive "anything (number, expression, whatever) raised to the power of zero has a value of 1"

Roger

Hopefully, this might make exponents a little clearer...

Thanks guys. Where were you when I was doing homework?

I was quite pleased with this 'show by example' as it's similar to Gary's more general approach

I would never have thought of John's logs in a month of Sundays. If only I'd paid attention at school!

Thanks,

Dave

I think I may have found my introductory text: **LINK**

http://mathinsight.org/interactive_gallery_quadric_surfaces_introduction

[One page in a very thoroughly hyperlinked collection of documents]

The interactive examples are written in Java; so won't run on the iPad, but should be O.K. on the Mac.

... so; I can transcribe equations to QuickGraph on the iPad as I work through them

Thanks again to John & Roger for the very helpful notes.

MichaelG.

z=z+z+z =zzz