Wolframalpha inverse plus#

Wolfram|Alpha Notebook Edition expands the scope of the Wolfram Predictive Interface. To explore an area of interest in depth, you can easily query for examples: Wolfram Language SyntaxĪs a hybrid between Mathematica and Wolfram|Alpha, Wolfram|Alpha Notebook Edition also supports parsing of proper Wolfram Language syntax: Wolfram Predictive Interface Related Computations Inputs by holding SHIFT while pressing ENTER ( RET). Unlike at Wolfram|Alpha, the natural language input cell supports multiline

You can add details and assumptions to your query, such as telling the system to use real numbers only: Just like on Wolfram|Alpha, you can write your input in conversational English: You can evaluate input by pressing ENTER ( RET) within the cell, or by clicking the button: Standard Usage There are two solutions () because the 4th circle can be either internally or externally tangent to the other three.Start typing in a blank notebook to generate a new input cell: R 4 - 2r 3 - 2r 2 - 2r + 1 = (r 2 - kr + 1)Ĭhoosing the positive root, and leaving the negative root for the anally retentive to write a song about, we have So we should expect a factor of the form r 2 - kr + 1, where k = r + (1/r), thus getting To paraphrase Snoddy, 'The sum of the squares of the radii equals half the square of their sum.'Īlgebraically, (1 + r + r 2 + r 3) 2 = 2(1 + r 2 + r 4 + r 6)īecause the GP is either increasing or decreasing, if r is a solution then 1/r is also a solution. Here is a painless method for solving the polynomial.

On 08/23/10 at 14:03:45, TenaliRaman wrote: That is phi + sqrt(phi) right? Post by ThudanBlunder on Aug 23 rd, 2010, 3:53pm We can use that if x is a solution, then so is 1/xĮxpand (x-a)(x-1/a)(x-b)(x-1/b), the coefficients have to be equal to the ones of x 4 - 2 x 3 - 2 x 2 - 2 x + 1, so we getįor real a,b, we'd have |a'|,|b'| >= 2, so we only need to look at We can dismiss (x 2 + 1) as a source for real solutions, so then we haveĪnd then I'm stuck letting wolframalpha finishing it off.

Wolframalpha inverse plus#

The Soddy circles formula plus geometric constraint givesĢ (x 6 + x 4 + x 2 + 1) = (x 3 + x 2 + x + 1) 2 But I still don't see a nice way to derive it. That is phi + sqrt(phi) right? It's beautiful! Post by TenaliRaman on Aug 23 rd, 2010, 2:03pm Wolframalpha is happy enough to solve it (giving 1/2+sqrt(5)/2+sqrt(1/2 (1+sqrt(5))) and it's inverse), but I'll think a bit more about whether there's a nice way to do it by hand. Then you only need to impose the constraints they're in geometric progression which gives you a six degree polynomial to solve. You can use the formula at () which gives the radius of 4 touching circles. If the radii of four mutually kissing circles are in geometric progression, find exact possible values for the common ratio. Post by ThudanBlunder on Aug 23 rd, 2010, 5:00am (Message started by: ThudanBlunder on Aug 23 rd, 2010, 5:00am) Wu :: forums - Print Page wu :: forums ()