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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 --- End quote --- 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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