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tech / sci.math / Re: Is there an equation for this?

SubjectAuthor
* Is there an equation for this?Dan joyce
+* Re: Is there an equation for this?sergio
|`* Re: Is there an equation for this?Dan joyce
| `* Re: Is there an equation for this?sergio
|  `- Re: Is there an equation for this?Dan joyce
`* Re: Is there an equation for this?Barry Schwarz
 `* Re: Is there an equation for this?Dan joyce
  `* Re: Is there an equation for this?Dan joyce
   +- Re: Is there an equation for this?Chris M. Thomasson
   +* Re: Is there an equation for this?Chris M. Thomasson
   |+* Re: Is there an equation for this?FromTheRafters
   ||`- Re: Is there an equation for this?sergio
   |`- Re: Is there an equation for this?Dan joyce
   +* Re: Is there an equation for this?Barry Schwarz
   |`- Re: Is there an equation for this?sergio
   `* Re: Is there an equation for this?Jim Burns
    `* Re: Is there an equation for this?Dan joyce
     `* Re: Is there an equation for this?sergio
      +- Re: Is there an equation for this?Dan joyce
      `* Re: Is there an equation for this?Jim Burns
       +- Re: Is there an equation for this?Dan joyce
       +* Re: Is there an equation for this?Chris M. Thomasson
       |+- Re: Is there an equation for this?Dan joyce
       |+- Re: Is there an equation for this?sergio
       |`* Re: Is there an equation for this?Phil Carmody
       | `* Re: Is there an equation for this?Dan joyce
       |  `- Re: Is there an equation for this?Oscar Yoshinobu
       `* Re: Is there an equation for this?Mike Terry
        `* Re: Is there an equation for this?Dan joyce
         +* Re: Is there an equation for this?Barry Schwarz
         |`- Re: Is there an equation for this?Dan joyce
         `* Re: Is there an equation for this?Mike Terry
          `* Re: Is there an equation for this?Dan joyce
           +* Re: Is there an equation for this?Mike Terry
           |`- Re: Is there an equation for this?Dan joyce
           `* Re: Is there an equation for this?Barry Schwarz
            `- Re: Is there an equation for this?Dan joyce

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Re: Is there an equation for this?

<f6deecaf-15f8-4979-ae06-56ee52666a1en@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=100014&group=sci.math#100014

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Subject: Re: Is there an equation for this?
From: danj4...@gmail.com (Dan joyce)
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 by: Dan joyce - Sat, 14 May 2022 04:34 UTC

On Friday, May 13, 2022 at 6:39:09 PM UTC-4, Mike Terry wrote:
> On 13/05/2022 19:03, Jim Burns wrote:
> > On 5/13/2022 11:35 AM, sergio wrote:
> >> On 5/13/2022 10:09 AM, Dan joyce wrote:
> >>> On Thursday, May 12, 2022 at 6:50:16 PM UTC-4,
> >>> Jim Burns wrote:
> >>>> On 5/12/2022 3:16 PM, Dan joyce wrote:
> >
> >>>>> Still trying to plug into Wolfram alpha
> >>>>> a formula with two known values Length (e)
> >>>>> and volume (pi) and inner space diagonal (pi).
> >>>>> To get width an height.
> >>>>> Anyone?
> >>>>
> >>>> pi = e*w*h, pi^2 = e^2+w^2+h^2
> >>>>
> >>>> One line, comma separated.
> >>>>
> >>>> I tried it, it works.
> >>>> There's an "exact form" button, if you'd like that.
> >>>
> >>> That is neat.
> >>> Thanks
> >
> >> wow! that is neat!! has 4 roots
> >
> > Agreed, very neat,
> > not that I helped build Wolfram Alpha or anything.
> >
> > The 4 roots are trivial variations of each other.
> > h₀=0.888927..., w₀=1.30014...
> >
> > h=h₀, w=w₀
> > h=-h₀, w=-w₀
> > h=w₀, w=h₀
> > h=-w₀, w=-h₀
> >
> >> and exact solutions are very complex,
> >
> > The good news is that Wolfram Alpha doesn't care
> > about "complex".
> >
> > The bad news is that Wolfram Alpha doesn't care
> > about "complex".
> >
> > We mere humans can do better than that, though.
> >
> > pi = e*w*h, pi^2 = e^2+w^2+h^2
> >
> > pi^2 = e^2*w^2*h^2
> >
> > W = w^2, H = h^2
> >
> > pi^2/e^2 = W*H, pi^2 - e^2 = W + H
> alternatively.. we have the WH and W+H (constant) values, so straight away W and H are the two roots
> of the quadratic
>
> x^2 - (W+H)x + (WH) = 0
>
> i.e. x^2 - (pi^2 - e^2)x + pi^2/e^2 = 0
>
> and we can use our favourite quadratic formula! (This way it's implicit that w,h can be swapped -
> they are the two roots, but either way round...)
>
> (Or we could find constants for wh, w+h directly from the start equations and get their quadratic
> directly, without introducing W,H)
>
> Mike.
> >
> > (pi^2-e^2)*H = W*H + H^2
> >
> > (pi^2-e^2)*H = (pi^2/e^2) + H^2
> >
> > H/(pi^2-e^2) = (pi^2/e^2)/(pi^2-e^2)^2 + (H/(pi^2-e^2))^2
> >
> > η = H/(pi^2-e^2)
> >
> > β = (pi^2/e^2)/(pi^2-e^2)^2
> >
> > η = β + η^2
> >
> > η = (1 - sqrt(1 - 4*β))/2
> > η = (1 + sqrt(1 - 4*β))/2
> >
> > h₀ = sqrt((pi^2-e^2)*(1 - sqrt(1 - 4*β))/2)
> > w₀ = sqrt((pi^2-e^2)*(1 + sqrt(1 - 4*β))/2)
> >
Mike,
Here is a different one I entered on Wolfram alpha but this time using the golden ratio
phi ---- e = phi*w*h, e^2 = phi^2+w^2+h^2
replacing pi with e and e with phi.
Don't get this one at all.
Complex, for sure.

Re: Is there an equation for this?

<65iu7hl0mf83g27s330oojc3aitl7ph2h2@4ax.com>

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https://www.novabbs.com/tech/article-flat.php?id=100020&group=sci.math#100020

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Subject: Re: Is there an equation for this?
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 by: Barry Schwarz - Sat, 14 May 2022 06:29 UTC

On Fri, 13 May 2022 21:34:34 -0700 (PDT), Dan joyce
<danj4084@gmail.com> wrote:

>On Friday, May 13, 2022 at 6:39:09 PM UTC-4, Mike Terry wrote:
>> On 13/05/2022 19:03, Jim Burns wrote:
>> > On 5/13/2022 11:35 AM, sergio wrote:
>> >> On 5/13/2022 10:09 AM, Dan joyce wrote:
>> >>> On Thursday, May 12, 2022 at 6:50:16 PM UTC-4,
>> >>> Jim Burns wrote:
>> >>>> On 5/12/2022 3:16 PM, Dan joyce wrote:
>> >
>> >>>>> Still trying to plug into Wolfram alpha
>> >>>>> a formula with two known values Length (e)
>> >>>>> and volume (pi) and inner space diagonal (pi).
>> >>>>> To get width an height.
>> >>>>> Anyone?
>> >>>>
>> >>>> pi = e*w*h, pi^2 = e^2+w^2+h^2
>> >>>>
>> >>>> One line, comma separated.
>> >>>>
>> >>>> I tried it, it works.
>> >>>> There's an "exact form" button, if you'd like that.
>> >>>
>> >>> That is neat.
>> >>> Thanks
>> >
>> >> wow! that is neat!! has 4 roots
>> >
>> > Agreed, very neat,
>> > not that I helped build Wolfram Alpha or anything.
>> >
>> > The 4 roots are trivial variations of each other.
>> > h?=0.888927..., w?=1.30014...
>> >
>> > h=h?, w=w?
>> > h=-h?, w=-w?
>> > h=w?, w=h?
>> > h=-w?, w=-h?
>> >
>> >> and exact solutions are very complex,
>> >
>> > The good news is that Wolfram Alpha doesn't care
>> > about "complex".
>> >
>> > The bad news is that Wolfram Alpha doesn't care
>> > about "complex".
>> >
>> > We mere humans can do better than that, though.
>> >
>> > pi = e*w*h, pi^2 = e^2+w^2+h^2
>> >
>> > pi^2 = e^2*w^2*h^2
>> >
>> > W = w^2, H = h^2
>> >
>> > pi^2/e^2 = W*H, pi^2 - e^2 = W + H
>> alternatively.. we have the WH and W+H (constant) values, so straight away W and H are the two roots
>> of the quadratic
>>
>> x^2 - (W+H)x + (WH) = 0
>>
>> i.e. x^2 - (pi^2 - e^2)x + pi^2/e^2 = 0
>>
>> and we can use our favourite quadratic formula! (This way it's implicit that w,h can be swapped -
>> they are the two roots, but either way round...)
>>
>> (Or we could find constants for wh, w+h directly from the start equations and get their quadratic
>> directly, without introducing W,H)
>>
>> Mike.
>> >
>> > (pi^2-e^2)*H = W*H + H^2
>> >
>> > (pi^2-e^2)*H = (pi^2/e^2) + H^2
>> >
>> > H/(pi^2-e^2) = (pi^2/e^2)/(pi^2-e^2)^2 + (H/(pi^2-e^2))^2
>> >
>> > ? = H/(pi^2-e^2)
>> >
>> > ? = (pi^2/e^2)/(pi^2-e^2)^2
>> >
>> > ? = ? + ?^2
>> >
>> > ? = (1 - sqrt(1 - 4*?))/2
>> > ? = (1 + sqrt(1 - 4*?))/2
>> >
>> > h? = sqrt((pi^2-e^2)*(1 - sqrt(1 - 4*?))/2)
>> > w? = sqrt((pi^2-e^2)*(1 + sqrt(1 - 4*?))/2)
>> >
>Mike,
>Here is a different one I entered on Wolfram alpha but this time using the golden ratio
>phi ---- e = phi*w*h, e^2 = phi^2+w^2+h^2
>replacing pi with e and e with phi.
>Don't get this one at all.
>Complex, for sure.

Complicated perhaps (in the eye of the beholder) but definitely real
and not complex.

The same analysis with the new values shows that the two missing sides
have lengths of 2.019682 and 0.831809.

These are the values produced by the exact expression
sqrt((e^2-phi^2 +/- sqrt((phi^2-e^2)^2 - 4*e^2/phi^2))/2)

If you call the value shared by the volume and diagonal k1 and the
length of the known side k2, then the general solution is
sqrt( (k1^2-k2^2 +/- sqrt((k2^2-k1^2)^2-4*k1^2/k2^2))/2 )

--
Remove del for email

Re: Is there an equation for this?

<efe2e5d3-b8c3-44cb-adfc-202c1ecf6962n@googlegroups.com>

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https://www.novabbs.com/tech/article-flat.php?id=100033&group=sci.math#100033

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Subject: Re: Is there an equation for this?
From: danj4...@gmail.com (Dan joyce)
Injection-Date: Sat, 14 May 2022 12:51:44 +0000
Content-Type: text/plain; charset="UTF-8"
 by: Dan joyce - Sat, 14 May 2022 12:51 UTC

On Saturday, May 14, 2022 at 2:29:55 AM UTC-4, Barry Schwarz wrote:
> On Fri, 13 May 2022 21:34:34 -0700 (PDT), Dan joyce
> <danj...@gmail.com> wrote:
>
> >On Friday, May 13, 2022 at 6:39:09 PM UTC-4, Mike Terry wrote:
> >> On 13/05/2022 19:03, Jim Burns wrote:
> >> > On 5/13/2022 11:35 AM, sergio wrote:
> >> >> On 5/13/2022 10:09 AM, Dan joyce wrote:
> >> >>> On Thursday, May 12, 2022 at 6:50:16 PM UTC-4,
> >> >>> Jim Burns wrote:
> >> >>>> On 5/12/2022 3:16 PM, Dan joyce wrote:
> >> >
> >> >>>>> Still trying to plug into Wolfram alpha
> >> >>>>> a formula with two known values Length (e)
> >> >>>>> and volume (pi) and inner space diagonal (pi).
> >> >>>>> To get width an height.
> >> >>>>> Anyone?
> >> >>>>
> >> >>>> pi = e*w*h, pi^2 = e^2+w^2+h^2
> >> >>>>
> >> >>>> One line, comma separated.
> >> >>>>
> >> >>>> I tried it, it works.
> >> >>>> There's an "exact form" button, if you'd like that.
> >> >>>
> >> >>> That is neat.
> >> >>> Thanks
> >> >
> >> >> wow! that is neat!! has 4 roots
> >> >
> >> > Agreed, very neat,
> >> > not that I helped build Wolfram Alpha or anything.
> >> >
> >> > The 4 roots are trivial variations of each other.
> >> > h?=0.888927..., w?=1.30014...
> >> >
> >> > h=h?, w=w?
> >> > h=-h?, w=-w?
> >> > h=w?, w=h?
> >> > h=-w?, w=-h?
> >> >
> >> >> and exact solutions are very complex,
> >> >
> >> > The good news is that Wolfram Alpha doesn't care
> >> > about "complex".
> >> >
> >> > The bad news is that Wolfram Alpha doesn't care
> >> > about "complex".
> >> >
> >> > We mere humans can do better than that, though.
> >> >
> >> > pi = e*w*h, pi^2 = e^2+w^2+h^2
> >> >
> >> > pi^2 = e^2*w^2*h^2
> >> >
> >> > W = w^2, H = h^2
> >> >
> >> > pi^2/e^2 = W*H, pi^2 - e^2 = W + H
> >> alternatively.. we have the WH and W+H (constant) values, so straight away W and H are the two roots
> >> of the quadratic
> >>
> >> x^2 - (W+H)x + (WH) = 0
> >>
> >> i.e. x^2 - (pi^2 - e^2)x + pi^2/e^2 = 0
> >>
> >> and we can use our favourite quadratic formula! (This way it's implicit that w,h can be swapped -
> >> they are the two roots, but either way round...)
> >>
> >> (Or we could find constants for wh, w+h directly from the start equations and get their quadratic
> >> directly, without introducing W,H)
> >>
> >> Mike.
> >> >
> >> > (pi^2-e^2)*H = W*H + H^2
> >> >
> >> > (pi^2-e^2)*H = (pi^2/e^2) + H^2
> >> >
> >> > H/(pi^2-e^2) = (pi^2/e^2)/(pi^2-e^2)^2 + (H/(pi^2-e^2))^2
> >> >
> >> > ? = H/(pi^2-e^2)
> >> >
> >> > ? = (pi^2/e^2)/(pi^2-e^2)^2
> >> >
> >> > ? = ? + ?^2
> >> >
> >> > ? = (1 - sqrt(1 - 4*?))/2
> >> > ? = (1 + sqrt(1 - 4*?))/2
> >> >
> >> > h? = sqrt((pi^2-e^2)*(1 - sqrt(1 - 4*?))/2)
> >> > w? = sqrt((pi^2-e^2)*(1 + sqrt(1 - 4*?))/2)
> >> >
> >Mike,
> >Here is a different one I entered on Wolfram alpha but this time using the golden ratio
> >phi ---- e = phi*w*h, e^2 = phi^2+w^2+h^2
> >replacing pi with e and e with phi.
> >Don't get this one at all.
> >Complex, for sure.
> Complicated perhaps (in the eye of the beholder) but definitely real
> and not complex.
>
> The same analysis with the new values shows that the two missing sides
> have lengths of 2.019682 and 0.831809.
>
> These are the values produced by the exact expression
> sqrt((e^2-phi^2 +/- sqrt((phi^2-e^2)^2 - 4*e^2/phi^2))/2)
>
> If you call the value shared by the volume and diagonal k1 and the
> length of the known side k2, then the general solution is
> sqrt( (k1^2-k2^2 +/- sqrt((k2^2-k1^2)^2-4*k1^2/k2^2))/2 )
> --
> Remove del for email
Thanks Barry.

Re: Is there an equation for this?

<t5omgs$1kst$1@gioia.aioe.org>

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Subject: Re: Is there an equation for this?
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 by: Mike Terry - Sat, 14 May 2022 16:48 UTC

On 14/05/2022 05:34, Dan joyce wrote:
> On Friday, May 13, 2022 at 6:39:09 PM UTC-4, Mike Terry wrote:
>> On 13/05/2022 19:03, Jim Burns wrote:
>>> On 5/13/2022 11:35 AM, sergio wrote:
>>>> On 5/13/2022 10:09 AM, Dan joyce wrote:
>>>>> On Thursday, May 12, 2022 at 6:50:16 PM UTC-4,
>>>>> Jim Burns wrote:
>>>>>> On 5/12/2022 3:16 PM, Dan joyce wrote:
>>>
>>>>>>> Still trying to plug into Wolfram alpha
>>>>>>> a formula with two known values Length (e)
>>>>>>> and volume (pi) and inner space diagonal (pi).
>>>>>>> To get width an height.
>>>>>>> Anyone?
>>>>>>
>>>>>> pi = e*w*h, pi^2 = e^2+w^2+h^2
>>>>>>
>>>>>> One line, comma separated.
>>>>>>
>>>>>> I tried it, it works.
>>>>>> There's an "exact form" button, if you'd like that.
>>>>>
>>>>> That is neat.
>>>>> Thanks
>>>
>>>> wow! that is neat!! has 4 roots
>>>
>>> Agreed, very neat,
>>> not that I helped build Wolfram Alpha or anything.
>>>
>>> The 4 roots are trivial variations of each other.
>>> h₀=0.888927..., w₀=1.30014...
>>>
>>> h=h₀, w=w₀
>>> h=-h₀, w=-w₀
>>> h=w₀, w=h₀
>>> h=-w₀, w=-h₀
>>>
>>>> and exact solutions are very complex,
>>>
>>> The good news is that Wolfram Alpha doesn't care
>>> about "complex".
>>>
>>> The bad news is that Wolfram Alpha doesn't care
>>> about "complex".
>>>
>>> We mere humans can do better than that, though.
>>>
>>> pi = e*w*h, pi^2 = e^2+w^2+h^2
>>>
>>> pi^2 = e^2*w^2*h^2
>>>
>>> W = w^2, H = h^2
>>>
>>> pi^2/e^2 = W*H, pi^2 - e^2 = W + H
>> alternatively.. we have the WH and W+H (constant) values, so straight away W and H are the two roots
>> of the quadratic
>>
>> x^2 - (W+H)x + (WH) = 0
>>
>> i.e. x^2 - (pi^2 - e^2)x + pi^2/e^2 = 0
>>
>> and we can use our favourite quadratic formula! (This way it's implicit that w,h can be swapped -
>> they are the two roots, but either way round...)
>>
>> (Or we could find constants for wh, w+h directly from the start equations and get their quadratic
>> directly, without introducing W,H)
>>
>> Mike.
>>>
>>> (pi^2-e^2)*H = W*H + H^2
>>>
>>> (pi^2-e^2)*H = (pi^2/e^2) + H^2
>>>
>>> H/(pi^2-e^2) = (pi^2/e^2)/(pi^2-e^2)^2 + (H/(pi^2-e^2))^2
>>>
>>> η = H/(pi^2-e^2)
>>>
>>> β = (pi^2/e^2)/(pi^2-e^2)^2
>>>
>>> η = β + η^2
>>>
>>> η = (1 - sqrt(1 - 4*β))/2
>>> η = (1 + sqrt(1 - 4*β))/2
>>>
>>> h₀ = sqrt((pi^2-e^2)*(1 - sqrt(1 - 4*β))/2)
>>> w₀ = sqrt((pi^2-e^2)*(1 + sqrt(1 - 4*β))/2)
>>>
> Mike,
> Here is a different one I entered on Wolfram alpha but this time using the golden ratio
> phi ---- e = phi*w*h, e^2 = phi^2+w^2+h^2
> replacing pi with e and e with phi.
> Don't get this one at all.
> Complex, for sure.
>

So our equations are

e = φ*w*h, e^2 = φ^2 + w^2 + h^2 ?

It's similar - we have

wh = e/φ
(W+h)^2 = w^2 + h^2 + 2wh
= e^2 - φ^2 + 2e/φ
so w+h = sqrt(e^2 - φ^2 + 2e/φ) // note positive root only

w and h will be the roots of

x^2 - sqrt(e^2 - φ^2 + 2e/φ).x + e/φ = 0

a quadratic equation. Using the quadratic formula, the roots are:

[sqrt(e^2 - φ^2 + 2e/φ) +- sqrt(e^2 - φ^2 + 2e/φ - 4e/φ)] / 2

i.e. [sqrt(e^2 - φ^2 + 2e/φ) +- sqrt(e^2 - φ^2 - 2e/φ)] / 2

giving w,h as approx 2.01968, 0.83181

Mike.

Re: Is there an equation for this?

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Subject: Re: Is there an equation for this?
From: danj4...@gmail.com (Dan joyce)
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 by: Dan joyce - Sat, 14 May 2022 18:17 UTC

On Saturday, May 14, 2022 at 12:48:38 PM UTC-4, Mike Terry wrote:
> On 14/05/2022 05:34, Dan joyce wrote:
> > On Friday, May 13, 2022 at 6:39:09 PM UTC-4, Mike Terry wrote:
> >> On 13/05/2022 19:03, Jim Burns wrote:
> >>> On 5/13/2022 11:35 AM, sergio wrote:
> >>>> On 5/13/2022 10:09 AM, Dan joyce wrote:
> >>>>> On Thursday, May 12, 2022 at 6:50:16 PM UTC-4,
> >>>>> Jim Burns wrote:
> >>>>>> On 5/12/2022 3:16 PM, Dan joyce wrote:
> >>>
> >>>>>>> Still trying to plug into Wolfram alpha
> >>>>>>> a formula with two known values Length (e)
> >>>>>>> and volume (pi) and inner space diagonal (pi).
> >>>>>>> To get width an height.
> >>>>>>> Anyone?
> >>>>>>
> >>>>>> pi = e*w*h, pi^2 = e^2+w^2+h^2
> >>>>>>
> >>>>>> One line, comma separated.
> >>>>>>
> >>>>>> I tried it, it works.
> >>>>>> There's an "exact form" button, if you'd like that.
> >>>>>
> >>>>> That is neat.
> >>>>> Thanks
> >>>
> >>>> wow! that is neat!! has 4 roots
> >>>
> >>> Agreed, very neat,
> >>> not that I helped build Wolfram Alpha or anything.
> >>>
> >>> The 4 roots are trivial variations of each other.
> >>> h₀=0.888927..., w₀=1.30014...
> >>>
> >>> h=h₀, w=w₀
> >>> h=-h₀, w=-w₀
> >>> h=w₀, w=h₀
> >>> h=-w₀, w=-h₀
> >>>
> >>>> and exact solutions are very complex,
> >>>
> >>> The good news is that Wolfram Alpha doesn't care
> >>> about "complex".
> >>>
> >>> The bad news is that Wolfram Alpha doesn't care
> >>> about "complex".
> >>>
> >>> We mere humans can do better than that, though.
> >>>
> >>> pi = e*w*h, pi^2 = e^2+w^2+h^2
> >>>
> >>> pi^2 = e^2*w^2*h^2
> >>>
> >>> W = w^2, H = h^2
> >>>
> >>> pi^2/e^2 = W*H, pi^2 - e^2 = W + H
> >> alternatively.. we have the WH and W+H (constant) values, so straight away W and H are the two roots
> >> of the quadratic
> >>
> >> x^2 - (W+H)x + (WH) = 0
> >>
> >> i.e. x^2 - (pi^2 - e^2)x + pi^2/e^2 = 0
> >>
> >> and we can use our favourite quadratic formula! (This way it's implicit that w,h can be swapped -
> >> they are the two roots, but either way round...)
> >>
> >> (Or we could find constants for wh, w+h directly from the start equations and get their quadratic
> >> directly, without introducing W,H)
> >>
> >> Mike.
> >>>
> >>> (pi^2-e^2)*H = W*H + H^2
> >>>
> >>> (pi^2-e^2)*H = (pi^2/e^2) + H^2
> >>>
> >>> H/(pi^2-e^2) = (pi^2/e^2)/(pi^2-e^2)^2 + (H/(pi^2-e^2))^2
> >>>
> >>> η = H/(pi^2-e^2)
> >>>
> >>> β = (pi^2/e^2)/(pi^2-e^2)^2
> >>>
> >>> η = β + η^2
> >>>
> >>> η = (1 - sqrt(1 - 4*β))/2
> >>> η = (1 + sqrt(1 - 4*β))/2
> >>>
> >>> h₀ = sqrt((pi^2-e^2)*(1 - sqrt(1 - 4*β))/2)
> >>> w₀ = sqrt((pi^2-e^2)*(1 + sqrt(1 - 4*β))/2)
> >>>
> > Mike,
> > Here is a different one I entered on Wolfram alpha but this time using the golden ratio
> > phi ---- e = phi*w*h, e^2 = phi^2+w^2+h^2
> > replacing pi with e and e with phi.
> > Don't get this one at all.
> > Complex, for sure.
> >
> So our equations are
>
> - ?
>
> It's similar - we have
>
> wh = e/φ
> (W+h)^2 = w^2 + h^2 + 2wh
> = e^2 - φ^2 + 2e/φ
> so w+h = sqrt(e^2 - φ^2 + 2e/φ) // note positive root only
>
> w and h will be the roots of
>
> x^2 - sqrt(e^2 - φ^2 + 2e/φ).x + e/φ = 0
>
> a quadratic equation. Using the quadratic formula, the roots are:
>
> [sqrt(e^2 - φ^2 + 2e/φ) +- sqrt(e^2 - φ^2 + 2e/φ - 4e/φ)] / 2
>
> i.e. [sqrt(e^2 - φ^2 + 2e/φ) +- sqrt(e^2 - φ^2 - 2e/φ)] / 2
>
> giving w,h as approx 2.01968, 0.83181
>
> Mike.
Mike
Entering it in Wolfram ---- e = phi*w*h, e^2 = phi^2 + w^2 + h^2
It calculates in a different manner then pi=e*w*h,pi^2= e^2+w^2+h^2
Try in Wolfram --- e = phi*w*h, e^2 = phi^2 + w^2 + h^2
It does not give the proper amounts for h or w.

Re: Is there an equation for this?

<t5ot4t$hpe$1@gioia.aioe.org>

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https://www.novabbs.com/tech/article-flat.php?id=100055&group=sci.math#100055

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From: news.dea...@darjeeling.plus.com (Mike Terry)
Newsgroups: sci.math
Subject: Re: Is there an equation for this?
Date: Sat, 14 May 2022 19:41:33 +0100
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 by: Mike Terry - Sat, 14 May 2022 18:41 UTC

On 14/05/2022 19:17, Dan joyce wrote:
> On Saturday, May 14, 2022 at 12:48:38 PM UTC-4, Mike Terry wrote:
>> On 14/05/2022 05:34, Dan joyce wrote:
>>> On Friday, May 13, 2022 at 6:39:09 PM UTC-4, Mike Terry wrote:
>>>> On 13/05/2022 19:03, Jim Burns wrote:
>>>>> On 5/13/2022 11:35 AM, sergio wrote:
>>>>>> On 5/13/2022 10:09 AM, Dan joyce wrote:
>>>>>>> On Thursday, May 12, 2022 at 6:50:16 PM UTC-4,
>>>>>>> Jim Burns wrote:
>>>>>>>> On 5/12/2022 3:16 PM, Dan joyce wrote:
>>>>>
>>>>>>>>> Still trying to plug into Wolfram alpha
>>>>>>>>> a formula with two known values Length (e)
>>>>>>>>> and volume (pi) and inner space diagonal (pi).
>>>>>>>>> To get width an height.
>>>>>>>>> Anyone?
>>>>>>>>
>>>>>>>> pi = e*w*h, pi^2 = e^2+w^2+h^2
>>>>>>>>
>>>>>>>> One line, comma separated.
>>>>>>>>
>>>>>>>> I tried it, it works.
>>>>>>>> There's an "exact form" button, if you'd like that.
>>>>>>>
>>>>>>> That is neat.
>>>>>>> Thanks
>>>>>
>>>>>> wow! that is neat!! has 4 roots
>>>>>
>>>>> Agreed, very neat,
>>>>> not that I helped build Wolfram Alpha or anything.
>>>>>
>>>>> The 4 roots are trivial variations of each other.
>>>>> h₀=0.888927..., w₀=1.30014...
>>>>>
>>>>> h=h₀, w=w₀
>>>>> h=-h₀, w=-w₀
>>>>> h=w₀, w=h₀
>>>>> h=-w₀, w=-h₀
>>>>>
>>>>>> and exact solutions are very complex,
>>>>>
>>>>> The good news is that Wolfram Alpha doesn't care
>>>>> about "complex".
>>>>>
>>>>> The bad news is that Wolfram Alpha doesn't care
>>>>> about "complex".
>>>>>
>>>>> We mere humans can do better than that, though.
>>>>>
>>>>> pi = e*w*h, pi^2 = e^2+w^2+h^2
>>>>>
>>>>> pi^2 = e^2*w^2*h^2
>>>>>
>>>>> W = w^2, H = h^2
>>>>>
>>>>> pi^2/e^2 = W*H, pi^2 - e^2 = W + H
>>>> alternatively.. we have the WH and W+H (constant) values, so straight away W and H are the two roots
>>>> of the quadratic
>>>>
>>>> x^2 - (W+H)x + (WH) = 0
>>>>
>>>> i.e. x^2 - (pi^2 - e^2)x + pi^2/e^2 = 0
>>>>
>>>> and we can use our favourite quadratic formula! (This way it's implicit that w,h can be swapped -
>>>> they are the two roots, but either way round...)
>>>>
>>>> (Or we could find constants for wh, w+h directly from the start equations and get their quadratic
>>>> directly, without introducing W,H)
>>>>
>>>> Mike.
>>>>>
>>>>> (pi^2-e^2)*H = W*H + H^2
>>>>>
>>>>> (pi^2-e^2)*H = (pi^2/e^2) + H^2
>>>>>
>>>>> H/(pi^2-e^2) = (pi^2/e^2)/(pi^2-e^2)^2 + (H/(pi^2-e^2))^2
>>>>>
>>>>> η = H/(pi^2-e^2)
>>>>>
>>>>> β = (pi^2/e^2)/(pi^2-e^2)^2
>>>>>
>>>>> η = β + η^2
>>>>>
>>>>> η = (1 - sqrt(1 - 4*β))/2
>>>>> η = (1 + sqrt(1 - 4*β))/2
>>>>>
>>>>> h₀ = sqrt((pi^2-e^2)*(1 - sqrt(1 - 4*β))/2)
>>>>> w₀ = sqrt((pi^2-e^2)*(1 + sqrt(1 - 4*β))/2)
>>>>>
>>> Mike,
>>> Here is a different one I entered on Wolfram alpha but this time using the golden ratio
>>> phi ---- e = phi*w*h, e^2 = phi^2+w^2+h^2
>>> replacing pi with e and e with phi.
>>> Don't get this one at all.
>>> Complex, for sure.
>>>
>> So our equations are
>>
>> - ?
>>
>> It's similar - we have
>>
>> wh = e/φ
>> (W+h)^2 = w^2 + h^2 + 2wh
>> = e^2 - φ^2 + 2e/φ
>> so w+h = sqrt(e^2 - φ^2 + 2e/φ) // note positive root only
>>
>> w and h will be the roots of
>>
>> x^2 - sqrt(e^2 - φ^2 + 2e/φ).x + e/φ = 0
>>
>> a quadratic equation. Using the quadratic formula, the roots are:
>>
>> [sqrt(e^2 - φ^2 + 2e/φ) +- sqrt(e^2 - φ^2 + 2e/φ - 4e/φ)] / 2
>>
>> i.e. [sqrt(e^2 - φ^2 + 2e/φ) +- sqrt(e^2 - φ^2 - 2e/φ)] / 2
>>
>> giving w,h as approx 2.01968, 0.83181
>>
>> Mike.
> Mike
> Entering it in Wolfram ---- e = phi*w*h, e^2 = phi^2 + w^2 + h^2
> It calculates in a different manner then pi=e*w*h,pi^2= e^2+w^2+h^2
> Try in Wolfram --- e = phi*w*h, e^2 = phi^2 + w^2 + h^2
> It does not give the proper amounts for h or w.
>

I don't use Wolfram, but maybe it doesn't understand phi without help? When I try

phi = (1+sqrt(5))/2, e = phi*w*h, e^2 = phi^2 + w^2 + h^2

it gets the numerical values of w,h above, although its "exact form" solution is rather mysterious
:) That often happens with solutions in radicals as there are different ways to express the same
values.

Mike.

Re: Is there an equation for this?

<h1vv7hdupg3acqn1catepb1493me1fm6mg@4ax.com>

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https://www.novabbs.com/tech/article-flat.php?id=100056&group=sci.math#100056

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Subject: Re: Is there an equation for this?
Date: Sat, 14 May 2022 12:02:59 -0700
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 by: Barry Schwarz - Sat, 14 May 2022 19:02 UTC

On Sat, 14 May 2022 11:17:25 -0700 (PDT), Dan joyce
<danj4084@gmail.com> wrote:

>On Saturday, May 14, 2022 at 12:48:38 PM UTC-4, Mike Terry wrote:
>> On 14/05/2022 05:34, Dan joyce wrote:

snip

>> > Mike,
>> > Here is a different one I entered on Wolfram alpha but this time using the golden ratio
>> > phi ---- e = phi*w*h, e^2 = phi^2+w^2+h^2
>> > replacing pi with e and e with phi.
>> > Don't get this one at all.
>> > Complex, for sure.
>> >
>> So our equations are
>>
>> - ?
>>
>> It's similar - we have
>>
>> wh = e/?
>> (W+h)^2 = w^2 + h^2 + 2wh
>> = e^2 - ?^2 + 2e/?
>> so w+h = sqrt(e^2 - ?^2 + 2e/?) // note positive root only
>>
>> w and h will be the roots of
>>
>> x^2 - sqrt(e^2 - ?^2 + 2e/?).x + e/? = 0
>>
>> a quadratic equation. Using the quadratic formula, the roots are:
>>
>> [sqrt(e^2 - ?^2 + 2e/?) +- sqrt(e^2 - ?^2 + 2e/? - 4e/?)] / 2
>>
>> i.e. [sqrt(e^2 - ?^2 + 2e/?) +- sqrt(e^2 - ?^2 - 2e/?)] / 2
>>
>> giving w,h as approx 2.01968, 0.83181
>>
>> Mike.
>Mike
>Entering it in Wolfram ---- e = phi*w*h, e^2 = phi^2 + w^2 + h^2
>It calculates in a different manner then pi=e*w*h,pi^2= e^2+w^2+h^2
>Try in Wolfram --- e = phi*w*h, e^2 = phi^2 + w^2 + h^2
>It does not give the proper amounts for h or w.

Which raises the question:

Are you interested in learning how to solve these types or problems or
are you interested in learning how to use Wolfram?

--
Remove del for email

Re: Is there an equation for this?

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Subject: Re: Is there an equation for this?
From: danj4...@gmail.com (Dan joyce)
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 by: Dan joyce - Sun, 15 May 2022 00:11 UTC

On Saturday, May 14, 2022 at 3:03:12 PM UTC-4, Barry Schwarz wrote:
> On Sat, 14 May 2022 11:17:25 -0700 (PDT), Dan joyce
> <danj...@gmail.com> wrote:
>
> >On Saturday, May 14, 2022 at 12:48:38 PM UTC-4, Mike Terry wrote:
> >> On 14/05/2022 05:34, Dan joyce wrote:
> snip
> >> > Mike,
> >> > Here is a different one I entered on Wolfram alpha but this time using the golden ratio
> >> > phi ---- e = phi*w*h, e^2 = phi^2+w^2+h^2
> >> > replacing pi with e and e with phi.
> >> > Don't get this one at all.
> >> > Complex, for sure.
> >> >
> >> So our equations are
> >>
> >> - ?
> >>
> >> It's similar - we have
> >>
> >> wh = e/?
> >> (W+h)^2 = w^2 + h^2 + 2wh
> >> = e^2 - ?^2 + 2e/?
> >> so w+h = sqrt(e^2 - ?^2 + 2e/?) // note positive root only
> >>
> >> w and h will be the roots of
> >>
> >> x^2 - sqrt(e^2 - ?^2 + 2e/?).x + e/? = 0
> >>
> >> a quadratic equation. Using the quadratic formula, the roots are:
> >>
> >> [sqrt(e^2 - ?^2 + 2e/?) +- sqrt(e^2 - ?^2 + 2e/? - 4e/?)] / 2
> >>
> >> i.e. [sqrt(e^2 - ?^2 + 2e/?) +- sqrt(e^2 - ?^2 - 2e/?)] / 2
> >>
> >> giving w,h as approx 2.01968, 0.83181
> >>
> >> Mike.
> >Mike
> >Entering it in Wolfram ---- e = phi*w*h, e^2 = phi^2 + w^2 + h^2
> >It calculates in a different manner then pi=e*w*h,pi^2= e^2+w^2+h^2
> >Try in Wolfram --- e = phi*w*h, e^2 = phi^2 + w^2 + h^2 > >It does not give the proper amounts for h or w.
> Which raises the question:
>
> Are you interested in learning how to solve these types or problems or
> are you interested in learning how to use Wolfram?
> --
> Remove del for email
No, I am just wondering why it gives a different resulting equation when the
amounts are switched e for pi and phi for e
e = phi*w*h, e^2 = phi^2 + w^2 + h^2
It does not give the correct answer.

Re: Is there an equation for this?

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Subject: Re: Is there an equation for this?
From: danj4...@gmail.com (Dan joyce)
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 by: Dan joyce - Sun, 15 May 2022 00:13 UTC

On Saturday, May 14, 2022 at 2:41:43 PM UTC-4, Mike Terry wrote:
> On 14/05/2022 19:17, Dan joyce wrote:
> > On Saturday, May 14, 2022 at 12:48:38 PM UTC-4, Mike Terry wrote:
> >> On 14/05/2022 05:34, Dan joyce wrote:
> >>> On Friday, May 13, 2022 at 6:39:09 PM UTC-4, Mike Terry wrote:
> >>>> On 13/05/2022 19:03, Jim Burns wrote:
> >>>>> On 5/13/2022 11:35 AM, sergio wrote:
> >>>>>> On 5/13/2022 10:09 AM, Dan joyce wrote:
> >>>>>>> On Thursday, May 12, 2022 at 6:50:16 PM UTC-4,
> >>>>>>> Jim Burns wrote:
> >>>>>>>> On 5/12/2022 3:16 PM, Dan joyce wrote:
> >>>>>
> >>>>>>>>> Still trying to plug into Wolfram alpha
> >>>>>>>>> a formula with two known values Length (e)
> >>>>>>>>> and volume (pi) and inner space diagonal (pi).
> >>>>>>>>> To get width an height.
> >>>>>>>>> Anyone?
> >>>>>>>>
> >>>>>>>> pi = e*w*h, pi^2 = e^2+w^2+h^2
> >>>>>>>>
> >>>>>>>> One line, comma separated.
> >>>>>>>>
> >>>>>>>> I tried it, it works.
> >>>>>>>> There's an "exact form" button, if you'd like that.
> >>>>>>>
> >>>>>>> That is neat.
> >>>>>>> Thanks
> >>>>>
> >>>>>> wow! that is neat!! has 4 roots
> >>>>>
> >>>>> Agreed, very neat,
> >>>>> not that I helped build Wolfram Alpha or anything.
> >>>>>
> >>>>> The 4 roots are trivial variations of each other.
> >>>>> h₀=0.888927..., w₀=1.30014...
> >>>>>
> >>>>> h=h₀, w=w₀
> >>>>> h=-h₀, w=-w₀
> >>>>> h=w₀, w=h₀
> >>>>> h=-w₀, w=-h₀
> >>>>>
> >>>>>> and exact solutions are very complex,
> >>>>>
> >>>>> The good news is that Wolfram Alpha doesn't care
> >>>>> about "complex".
> >>>>>
> >>>>> The bad news is that Wolfram Alpha doesn't care
> >>>>> about "complex".
> >>>>>
> >>>>> We mere humans can do better than that, though.
> >>>>>
> >>>>> pi = e*w*h, pi^2 = e^2+w^2+h^2
> >>>>>
> >>>>> pi^2 = e^2*w^2*h^2
> >>>>>
> >>>>> W = w^2, H = h^2
> >>>>>
> >>>>> pi^2/e^2 = W*H, pi^2 - e^2 = W + H
> >>>> alternatively.. we have the WH and W+H (constant) values, so straight away W and H are the two roots
> >>>> of the quadratic
> >>>>
> >>>> x^2 - (W+H)x + (WH) = 0
> >>>>
> >>>> i.e. x^2 - (pi^2 - e^2)x + pi^2/e^2 = 0
> >>>>
> >>>> and we can use our favourite quadratic formula! (This way it's implicit that w,h can be swapped -
> >>>> they are the two roots, but either way round...)
> >>>>
> >>>> (Or we could find constants for wh, w+h directly from the start equations and get their quadratic
> >>>> directly, without introducing W,H)
> >>>>
> >>>> Mike.
> >>>>>
> >>>>> (pi^2-e^2)*H = W*H + H^2
> >>>>>
> >>>>> (pi^2-e^2)*H = (pi^2/e^2) + H^2
> >>>>>
> >>>>> H/(pi^2-e^2) = (pi^2/e^2)/(pi^2-e^2)^2 + (H/(pi^2-e^2))^2
> >>>>>
> >>>>> η = H/(pi^2-e^2)
> >>>>>
> >>>>> β = (pi^2/e^2)/(pi^2-e^2)^2
> >>>>>
> >>>>> η = β + η^2
> >>>>>
> >>>>> η = (1 - sqrt(1 - 4*β))/2
> >>>>> η = (1 + sqrt(1 - 4*β))/2
> >>>>>
> >>>>> h₀ = sqrt((pi^2-e^2)*(1 - sqrt(1 - 4*β))/2)
> >>>>> w₀ = sqrt((pi^2-e^2)*(1 + sqrt(1 - 4*β))/2)
> >>>>>
> >>> Mike,
> >>> Here is a different one I entered on Wolfram alpha but this time using the golden ratio
> >>> phi ---- e = phi*w*h, e^2 = phi^2+w^2+h^2
> >>> replacing pi with e and e with phi.
> >>> Don't get this one at all.
> >>> Complex, for sure.
> >>>
> >> So our equations are
> >>
> >> - ?
> >>
> >> It's similar - we have
> >>
> >> wh = e/φ
> >> (W+h)^2 = w^2 + h^2 + 2wh
> >> = e^2 - φ^2 + 2e/φ
> >> so w+h = sqrt(e^2 - φ^2 + 2e/φ) // note positive root only
> >>
> >> w and h will be the roots of
> >>
> >> x^2 - sqrt(e^2 - φ^2 + 2e/φ).x + e/φ = 0
> >>
> >> a quadratic equation. Using the quadratic formula, the roots are:
> >>
> >> [sqrt(e^2 - φ^2 + 2e/φ) +- sqrt(e^2 - φ^2 + 2e/φ - 4e/φ)] / 2
> >>
> >> i.e. [sqrt(e^2 - φ^2 + 2e/φ) +- sqrt(e^2 - φ^2 - 2e/φ)] / 2
> >>
> >> giving w,h as approx 2.01968, 0.83181
> >>
> >> Mike.
> > Mike
> > Entering it in Wolfram ---- e = phi*w*h, e^2 = phi^2 + w^2 + h^2
> > It calculates in a different manner then pi=e*w*h,pi^2= e^2+w^2+h^2
> > Try in Wolfram --- e = phi*w*h, e^2 = phi^2 + w^2 + h^2
> > It does not give the proper amounts for h or w.
> >
> I don't use Wolfram, but maybe it doesn't understand phi without help? When I try
>
> phi = (1+sqrt(5))/2, e = phi*w*h, e^2 = phi^2 + w^2 + h^2
>
> it gets the numerical values of w,h above, although its "exact form" solution is rather mysterious
> :) That often happens with solutions in radicals as there are different ways to express the same
> values.
>
> Mike.
Thanks Mike

Re: Is there an equation for this?

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From: pc+use...@asdf.org (Phil Carmody)
Newsgroups: sci.math
Subject: Re: Is there an equation for this?
Date: Sun, 15 May 2022 21:42:20 +0300
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 by: Phil Carmody - Sun, 15 May 2022 18:42 UTC

"Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> writes:
> The two square roots definitely remind me of choosing roots in a
> reverse iteration Julia set. Check this out, the main equations are
> from me:
>
> http://paulbourke.net/fractals/multijulia
>
> Paul was nice enough to experiment with it, and dedicate server space
> to show it.

Cool stuff - good work, both!

Phil
--
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gained some understanding of the world in which we live. As such, we can cast
aside childish remnants from the dawn of our civilization.
-- NotSanguine on SoylentNews, after Eugen Weber in /The Western Tradition/

Re: Is there an equation for this?

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Subject: Re: Is there an equation for this?
From: danj4...@gmail.com (Dan joyce)
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 by: Dan joyce - Sun, 15 May 2022 20:41 UTC

On Sunday, May 15, 2022 at 2:42:30 PM UTC-4, Phil Carmody wrote:
> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:
> > The two square roots definitely remind me of choosing roots in a
> > reverse iteration Julia set. Check this out, the main equations are
> > from me:
> >
> > http://paulbourke.net/fractals/multijulia
> >
> > Paul was nice enough to experiment with it, and dedicate server space
> > to show it.
> Cool stuff - good work, both!

Ditto!
Most all of Chris's work in these renderings are fascinating.
It truly is an art form in a branch of complex mathematics.

Dan

> Phil
> --
> We are no longer hunters and nomads. No longer awed and frightened, as we have
> gained some understanding of the world in which we live. As such, we can cast
> aside childish remnants from the dawn of our civilization.
> -- NotSanguine on SoylentNews, after Eugen Weber in /The Western Tradition/

Re: Is there an equation for this?

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From: bco...@inhycoou.uo (Oscar Yoshinobu)
Newsgroups: sci.math
Subject: Re: Is there an equation for this?
Date: Sun, 15 May 2022 21:00:21 -0000 (UTC)
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 by: Oscar Yoshinobu - Sun, 15 May 2022 21:00 UTC

Dan joyce wrote:

>> > http://paulbourke.net/fractals/multijulia
>> >
>> > Paul was nice enough to experiment with it, and dedicate server space
>> > to show it.
>> Cool stuff - good work, both!
>
> Ditto!
> Most all of Chris's work in these renderings are fascinating. It truly
> is an art form in a branch of complex mathematics.

fractals, idiot. I did this as 10 years old.

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