General Mathematics Discussion Thread

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tyteen4a03
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General Mathematics Discussion Thread

Post by tyteen4a03 » Fri Jul 06, 2012 8:08 pm

Blast!10 wrote:This topic is to discuss anything and everything math related. Talk about conversion formulas, fun facts, proof, bitwise operators, algebra, and everything else.
On WolframAlpha, they have a very cool graph drawer that draws the graph according to the function given. I googled around and found this inequality, which draws a batman. The inequality is this:

Code: Select all

x^2/49+y^2/9-1<=0 and abs&#40;x&#41;>=4 and -&#40;3 sqrt&#40;33&#41;&#41;/7<=y<=0 or abs&#40;x&#41;>=3 and y>=0 or -3<=y<=0 and -4<=x<=4 and &#40;abs&#40;x&#41;&#41;/2+sqrt&#40;1-&#40;abs&#40;abs&#40;x&#41;-2&#41;-1&#41;^2&#41;-1/112 &#40;3 sqrt&#40;33&#41;-7&#41; x^2-y-3<=0 or y>=0 and 3/4<=abs&#40;x&#41;<=1 and -8 abs&#40;x&#41;-y+9>=0 or 1/2<=abs&#40;x&#41;<=3/4 and 3 abs&#40;x&#41;-y+3/4>=0 and y>=0 or abs&#40;x&#41;<=1/2 and y>=0 and 9/4-y>=0 or abs&#40;x&#41;>=1 and y>=0 and -&#40;abs&#40;x&#41;&#41;/2-3/7 sqrt&#40;10&#41; sqrt&#40;4-&#40;abs&#40;x&#41;-1&#41;^2&#41;-y+&#40;6 sqrt&#40;10&#41;&#41;/7+3/2>=0
SVG images are vector images defined with functions just like these. While very time-consuming to get it right, it does look beautiful at the end.
and the duck went moo

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VirtLands
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mathematics topic

Post by VirtLands » Mon Jul 09, 2012 8:43 pm

[ I deleted my previous post here, since it was not advancing ..]

Hi, yes it's an addicting topic, here are related links>

Wolfram Alpha:
http://www.wolframalpha.com/#

Wolfram Mathworld:
http://mathworld.wolfram.com/

Wolfram Math Online Integrator:
http://integrals.wolfram.com/index.jsp


Some of my Wolfram favorites
:
_________________________________________________________
600-cell : hypericosahedron
http://en.wikipedia.org/wiki/600-cell
http://mathworld.wolfram.com/600-Cell.html
http://www.wolframalpha.com/input/?i=600-cell

120-cell : hecatonicosachoron
http://mathworld.wolfram.com/120-Cell.html
http://www.wolframalpha.com/input/?i=120+cell
_________________________________________________________
11-cell :
http://en.wikipedia.org/wiki/11-cell

16-cell : also known as hyperoctahedron or hexadecachoron

24-cell : also known as hyperdiamond or icositetrachoron
http://mathworld.wolfram.com/24-Cell.html
http://en.wikipedia.org/wiki/24-cell
_________________________________________________________
57-cell :
There are 57 vertices and each vertex connects to 6 others; I haven't
proved this to myself yet, so I'll take their word for it.

higher-resolution view at this link:
http://upload.wikimedia.org/wikipedia/c ... gs.svg.png
__________________________________________________________________________________________
Uniform polychorons: http://en.wikipedia.org/wiki/Uniform_po ... _H4_family

A stereographic preview of the 600-Cell:
Note: each vertex connects to 12 other vertexes.
Schläfli symbol {3,3,5}
It contains 1200 triangular faces, 720 edges, and 120 vertices. The edges form 72 flat regular decagons.
Each vertex of the 600-cell is a vertex of six such decagons.
http://en.wikipedia.org/wiki/File:Stere ... 00cell.png
Image
Last edited by VirtLands on Thu Dec 20, 2012 11:42 pm, edited 2 times in total.
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Post by llarson » Mon Jul 09, 2012 8:49 pm

I LOOOOVE geometry so I guess I'll post about my favorite shape. Introducing: THE ICOSAGON
In other words: It's a 20 sided shape. Looks almost like a circle somewhat. I looked it up on wikipedia one time and it almost looked exaclty like a circle. BTW Virtlands is there an 100-sided shape?




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Post by StinkerSquad01 » Mon Jul 09, 2012 9:18 pm

There is a shape for any number of sides. 100 sided shape is 100-gon or Hectagon.
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regular polyhedra, polygons, & polychorons

Post by VirtLands » Mon Jul 09, 2012 9:28 pm

llarson wrote:I LOOOOVE geometry so I guess I'll post about my favorite shape. Introducing: THE ICOSAGON

The IcosaGon: a 2-dimensional regular flat 20-sided shape:

http://mathworld.wolfram.com/Icosagon.html

Given a unit side length of u=1, you can accurately calculate the
inradius, circumradius, of an Icosagon as follows:

r = u * ½(1 + &#8730;5 + &#8730;(5 + 2&#8730;5)) : (inradius)
R = u * &#8730;(3 + &#8730;5 + ½&#8730;(50 + 22&#8730;5)) : (circumradius)
A = u² * 10(1 + &#8730;5 + &#8730;(5 + 2&#8730;5)) : (Area)
Last edited by VirtLands on Wed Jul 11, 2012 5:37 pm, edited 1 time in total.
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Post by llarson » Mon Jul 09, 2012 9:29 pm

WOW! :o THANKS A LOT!!!!!!!! :D :D :D Now where did that like button run off to?
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defined irrationals in trigonometry

Post by VirtLands » Mon Jul 09, 2012 11:16 pm

In Trigonometry, ...

Some of the irrational SIN and COS values are precisely defined,
and some are NOT:

For Examples:

NOT perfectly defined : cos(10°) &#8776; 0.9848077530
NOT perfectly defined : cos(11°) &#8776; 0.9816271834

IS perfectly defined : COS(15°) = ¼(&#8730;6+&#8730;2)
IS perfectly defined : COS(18.75°) = ¼&#8730; (8+2&#8730;(8+2(&#8730;6-&#8730;2)))
IS perfectly defined : SIN(45°) = &#8730;(0.5) &#8776; 0.7071067811

Unbelievable as it is, here are two more unusual trig cases that ARE precisely defined:

cos( 360/17° ) = [-1 + &#8730;17 + &#8730;(34-2&#8730;17) + 2&#8730;(17 +3&#8730;17 -&#8730;(34-2&#8730;17) -2&#8730;(34+2&#8730;17) )]/16

sin( 20° ) = 2 * 2^(&#8531;) * (³&#8730;(i-&#8730;3) -³&#8730;(i+&#8730;3)) :shock:

(where i=&#8730;-1, PI = 3.141592653589.., and ³&#8730; may represent the third root of a complex number)

You can find a few more examples like this on these links:

Trigonometry Angles--Pi/5 : http://mathworld.wolfram.com/TrigonometryAnglesPi5.html
Trigonometry Angles--Pi/7 : http://mathworld.wolfram.com/TrigonometryAnglesPi7.html
Trigonometry Angles--Pi/9 : http://mathworld.wolfram.com/TrigonometryAnglesPi9.html
Trigonometry Angles--Pi/11 : http://mathworld.wolfram.com/Trigonomet ... sPi11.html
Trigonometry Angles--Pi/13 : http://mathworld.wolfram.com/Trigonomet ... sPi13.html
Trigonometry Angles--Pi/15 : http://mathworld.wolfram.com/Trigonomet ... sPi15.html
Trigonometry Angles--Pi/17 : http://mathworld.wolfram.com/Trigonomet ... sPi17.html
Trigonometry Angles--Pi/23 : http://mathworld.wolfram.com/Trigonomet ... sPi23.html

Trigonometry Angles : http://mathworld.wolfram.com/TrigonometryAngles.html

Okay, so maybe since sin(20°) is defined, then also sin(10°)
can also be perfectly defined using half-angle formulas.

There is more on this, but it's over my head right now.
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Post by tyteen4a03 » Tue Jul 10, 2012 6:27 am

sin/cos/tan(30, 45, 60, 90) are also precisely defined, and are more common.

The ³&#8730; can also be represented with a power to 1/3.
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Post by dlcs18 » Sun Aug 05, 2012 2:02 pm

I figured out a formula which calculates the width/height of a rectangle if you have the area and the horizontal-to-vertical ratio.

(sqrt(a)*sqrt(b))/sqrt(c)

where:
a = the area of the rectangle
The values of b and c depend on if you want the width or the height of the region. If you want width:
b = the horizontal part of the ratio
c = the vertical part of the ratio
If you want height:
b = the vertical part of the ratio
c = the horizontal part of the ratio


For example, on a screen, a region of 640*480 pixels has the total pixel amount of 307200, and a ratio of 4:3
So if we insert these into the formula:
(sqrt(307200)*sqrt(4))/sqrt(3)
it will give us 640, which is the width of the region.
or to calculate the height, switch the 4 and 3 around:
(sqrt(307200)*sqrt(3))/sqrt(4)
which will give you 480, which is the height of the region.
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Finding W,H, given Area and ratio

Post by VirtLands » Mon Aug 06, 2012 6:54 pm

Hello dlcs18, my hat's off to you.: Here's my version of the same events:

A = Area : r = ratio : h = height : w = width
( r=4/3 , A = 307200 )

h * (&#8308;/&#8323; h = 307200
h² * (&#8308;/&#8323;) = 307200
h² = 307200 * (¾)
h² = 230400
h = 480
w = 480 * &#8308;/&#8323; = 640
________________________________________________________
or, creating the algorithm, where height (h) = 'the shorter side', and r= w/h
A = h²*r
H = &#8730;( A/r ), W = H*r
Last edited by VirtLands on Fri Dec 21, 2012 1:28 am, edited 1 time in total.
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collatz_conjecture

Post by VirtLands » Fri Dec 21, 2012 1:27 am

-- Image ----- Image

[ Doing stuff the hard way.. ]
-- Image ----- Image

-- Image

-- Image

-- Image

-- Image ----- Image

-- Image

-- Image

-- Image

[ All of the above comix were taken from:
http://xkcd.com/about/
http://xkcd.com/archive/
] :)

-------------------------------------------------------------------------------------
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L'Carpetron Dookmarriot 3
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Post by L'Carpetron Dookmarriot 3 » Fri Dec 21, 2012 4:14 am

Image

LOL! :lol: :lol: :lol: :lol:
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Post by llarson » Fri Dec 21, 2012 4:26 am

Solution wrote:Because the goat can't be alone with the cabbage :wink: .
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Post by L'Carpetron Dookmarriot 3 » Fri Dec 21, 2012 4:38 am

There is supposed to be a solution where you bring all of them over, (even the wolf) but he tells us to just ignore the wolf because well, why would you have a wolf?
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Post by Sammy_P » Fri Dec 21, 2012 5:04 am

L'Carpetron Dookmarriot 3 wrote:Image

LOL! :lol: :lol: :lol: :lol:
At this point, we need a topic for xkcd.
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