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The sometimes 'Beauty' of mathematics???

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AntiProtonBoy:
One particular construct I find super elegant in mathematics is transformation matrices used in visualising geometry and inverse kinematics. These matrices are almost like magic. You can combine a sequence of transform operations into a single matrix and get the correct result in a single operation.

Circlotron:
Can someone please explain how these cool mathematical principles came about as a consequence of a mindless universe beginning with a whopping big explosion? Or did they exist beforehand?

AntiProtonBoy:
Most mathematical principles is a form of "abstract logic" (for a lack of a better term), which may not need be based on the properties of the universe. Of course the universe needs to exist for us to have the mind to think 1+1=2, but the math alone does not have to define anything about the universe.

GlennSprigg:

--- Quote from: basinstreetdesign on August 09, 2021, 03:40:12 am ---
--- Quote from: The Electrician on August 05, 2021, 11:17:07 pm ---
--- Quote from: GlennSprigg on August 04, 2021, 01:48:17 pm ---I don't know how many people have seen my 'Tag' (?) thing, at the bottom of my posts/replies?   :P
   Diagonal of 1x1 square = Root-2. Ok.
   Diagonal of 1x1x1 cube = Root-3 !!!  Beautiful !!

Doesn't that not strike people as mathematically amazing! that the internal diagonal of a 'Cube' (1x1x1) = Root-3 !!!
I love the natural relationships with certain numbers, that can easily be proven/shown, and be so simple!   8)
GREAT!!  Good-ol basic maths can show/prove it all !!!...

Then I found out recently, that there is NO KNOWN correct formula, to calculate the Circumference of an 'Ellipse' !!   :palm:
You don't believe it... then look it up!!  Sigh...   Just lost faith in Maths again...   :box:

--- End quote ---

Of course there is a correct formula.  The relevant observation is found on this page: http://www.numericana.com/answer/ellipse.htm#elliptic

"There is no simple exact formula:  There are simple formulas but they are not exact, and there are exact formulas but they are not simple."

Also here: https://wj32.org/wp/2012/12/15/formula-for-the-circumference-of-an-ellipse/

--- End quote ---

I'm surprised that no one has mentioned that a simple extension of your sequence is that the "diagonal" (space-diagonal) of a 1 x 1 x 1 x 1 hypercube or tesseract is root(4) = 2

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Ah!! Yes... I've thought of that... and although 'logical', it is some-what impossible to 'imagine' such a 'diagonal'.  Here is a Mental 'representation' of
a HyperCube/Tesseract, as it can be perceived in 3-Dimesions in our mind, before going on here...

--- End quote ---

A 'square' has 4 right-angles. a 'Cube' has all right-angles too!, but we 'draw' one with distorted angles to represent it in 3-D.  A 'HyperCube' consists
of technically a Cube within a Cube, with lines joining the cubes too, and even THESE are at right-angles to everything else within!  Of course, our
brains can't fathom/picture that, so like with the 'cube' drawing, we can only imagine it as a 3-D 'shadow, and seeing distorted angles, as above!
We can't even IMAGINE where to place such a 'diagonal', within such an elusive concept!!!    :palm:

P.S.   I haven't finished with orig/prior discussions yet, though !!!   8)

tggzzz:
At this point, Flatland: A Romance of Many Dimensions by George Abbott is relevant.

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