In this geometric mandala, there are 7 constituent shapes and we can
use circles in corresponding colors to visualize the area of these
shapes (ordering them from the smallest at the top to larger ones
towards the bottom):

https://www.desmos.com/calculator/2pnzxwil3y

If we graph the length of the outline of each shape as a function
of the area it encloses, we can see how the shapes compare
to their associated circles in the way that they deviate from
optimal (in the sense of circles having the minimal length of outline
that is able to enclose a given amount of area):

https://www.desmos.com/calculator/i0n66pmccn

We can also morph between a particular shape and its associated
circle, where the amount of area enclosed remains constant as
the morphing outline increases in length (as it deviates from the
optimal circle shape):

https://www.desmos.com/calculator/zgj65ahcdi

Is there any math wizard here that has some ideas of how to
obtain a similar morphing for the other shapes?
What would be the most suitable way to accomplish this in
desmos?

On Monday, July 26, 2021 at 2:17:08 AM UTC+2, sobriquet wrote:
> In this geometric mandala, there are 7 constituent shapes and we can
> use circles in corresponding colors to visualize the area of these
> shapes (ordering them from the smallest at the top to larger ones
> towards the bottom):
>
> https://www.desmos.com/calculator/2pnzxwil3y
>
> If we graph the length of the outline of each shape as a function
> of the area it encloses, we can see how the shapes compare
> to their associated circles in the way that they deviate from
> optimal (in the sense of circles having the minimal length of outline
> that is able to enclose a given amount of area):
>
> https://www.desmos.com/calculator/i0n66pmccn
>
> We can also morph between a particular shape and its associated
> circle, where the amount of area enclosed remains constant as
> the morphing outline increases in length (as it deviates from the
> optimal circle shape):
>
> https://www.desmos.com/calculator/zgj65ahcdi
>
> Is there any math wizard here that has some ideas of how to
> obtain a similar morphing for the other shapes?
> What would be the most suitable way to accomplish this in
> desmos?

Funny that the same shapes crop up in other unexpected places, like this
interesting Desmos demo I found on Reddit related to automorphic forms.

https://i.imgur.com/Qy6RxiN.png

https://www.desmos.com/calculator/ap72luvpdo

On Saturday, July 31, 2021 at 11:44:05 AM UTC-7, sobriquet wrote:
> On Monday, July 26, 2021 at 2:17:08 AM UTC+2, sobriquet wrote:
> > In this geometric mandala, there are 7 constituent shapes and we can
> > use circles in corresponding colors to visualize the area of these
> > shapes (ordering them from the smallest at the top to larger ones
> > towards the bottom):
> >
> > https://www.desmos.com/calculator/2pnzxwil3y
> >
> > If we graph the length of the outline of each shape as a function
> > of the area it encloses, we can see how the shapes compare
> > to their associated circles in the way that they deviate from
> > optimal (in the sense of circles having the minimal length of outline
> > that is able to enclose a given amount of area):
> >
> > https://www.desmos.com/calculator/i0n66pmccn
> >
> > We can also morph between a particular shape and its associated
> > circle, where the amount of area enclosed remains constant as
> > the morphing outline increases in length (as it deviates from the
> > optimal circle shape):
> >
> > https://www.desmos.com/calculator/zgj65ahcdi
> >
> > Is there any math wizard here that has some ideas of how to
> > obtain a similar morphing for the other shapes?
> > What would be the most suitable way to accomplish this in
> > desmos?
> Funny that the same shapes crop up in other unexpected places, like this
> interesting Desmos demo I found on Reddit related to automorphic forms.
>
> https://i.imgur.com/Qy6RxiN.png
>
> https://www.desmos.com/calculator/ap72luvpdo

"Interpolation"

On Sunday, August 1, 2021 at 6:40:44 PM UTC+2, Ross A. Finlayson wrote:
> On Saturday, July 31, 2021 at 11:44:05 AM UTC-7, sobriquet wrote:
> > On Monday, July 26, 2021 at 2:17:08 AM UTC+2, sobriquet wrote:
> > > In this geometric mandala, there are 7 constituent shapes and we can
> > > use circles in corresponding colors to visualize the area of these
> > > shapes (ordering them from the smallest at the top to larger ones
> > > towards the bottom):
> > >
> > > https://www.desmos.com/calculator/2pnzxwil3y
> > >
> > > If we graph the length of the outline of each shape as a function
> > > of the area it encloses, we can see how the shapes compare
> > > to their associated circles in the way that they deviate from
> > > optimal (in the sense of circles having the minimal length of outline
> > > that is able to enclose a given amount of area):
> > >
> > > https://www.desmos.com/calculator/i0n66pmccn
> > >
> > > We can also morph between a particular shape and its associated
> > > circle, where the amount of area enclosed remains constant as
> > > the morphing outline increases in length (as it deviates from the
> > > optimal circle shape):
> > >
> > > https://www.desmos.com/calculator/zgj65ahcdi
> > >
> > > Is there any math wizard here that has some ideas of how to
> > > obtain a similar morphing for the other shapes?
> > > What would be the most suitable way to accomplish this in
> > > desmos?
> > Funny that the same shapes crop up in other unexpected places, like this
> > interesting Desmos demo I found on Reddit related to automorphic forms.
> >
> > https://i.imgur.com/Qy6RxiN.png
> >
> > https://www.desmos.com/calculator/ap72luvpdo
> "Interpolation"

Sure, but it gets tricky as the shapes get more irregular if you want to keep track
of what happens to the area of the shapes as you morph between them.

https://www.desmos.com/calculator/klj62wsw8l

On Tuesday, August 3, 2021 at 2:49:53 PM UTC+2, sobriquet wrote:
> On Sunday, August 1, 2021 at 6:40:44 PM UTC+2, Ross A. Finlayson wrote:
> > On Saturday, July 31, 2021 at 11:44:05 AM UTC-7, sobriquet wrote:
> > > On Monday, July 26, 2021 at 2:17:08 AM UTC+2, sobriquet wrote:
> > > > In this geometric mandala, there are 7 constituent shapes and we can
> > > > use circles in corresponding colors to visualize the area of these
> > > > shapes (ordering them from the smallest at the top to larger ones
> > > > towards the bottom):
> > > >
> > > > https://www.desmos.com/calculator/2pnzxwil3y
> > > >
> > > > If we graph the length of the outline of each shape as a function
> > > > of the area it encloses, we can see how the shapes compare
> > > > to their associated circles in the way that they deviate from
> > > > optimal (in the sense of circles having the minimal length of outline
> > > > that is able to enclose a given amount of area):
> > > >
> > > > https://www.desmos.com/calculator/i0n66pmccn
> > > >
> > > > We can also morph between a particular shape and its associated
> > > > circle, where the amount of area enclosed remains constant as
> > > > the morphing outline increases in length (as it deviates from the
> > > > optimal circle shape):
> > > >
> > > > https://www.desmos.com/calculator/zgj65ahcdi
> > > >
> > > > Is there any math wizard here that has some ideas of how to
> > > > obtain a similar morphing for the other shapes?
> > > > What would be the most suitable way to accomplish this in
> > > > desmos?
> > > Funny that the same shapes crop up in other unexpected places, like this
> > > interesting Desmos demo I found on Reddit related to automorphic forms.
> > >
> > > https://i.imgur.com/Qy6RxiN.png
> > >
> > > https://www.desmos.com/calculator/ap72luvpdo
> > "Interpolation"
> Sure, but it gets tricky as the shapes get more irregular if you want to keep track
> of what happens to the area of the shapes as you morph between them.
>
> https://www.desmos.com/calculator/klj62wsw8l

Maybe this course will yield some clues.

https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS

On 7/25/2021 5:17 PM, sobriquet wrote:
>
> In this geometric mandala, there are 7 constituent shapes and we can
> use circles in corresponding colors to visualize the area of these
> shapes (ordering them from the smallest at the top to larger ones
> towards the bottom):
>
> https://www.desmos.com/calculator/2pnzxwil3y
>
> If we graph the length of the outline of each shape as a function
> of the area it encloses, we can see how the shapes compare
> to their associated circles in the way that they deviate from
> optimal (in the sense of circles having the minimal length of outline
> that is able to enclose a given amount of area):
>
> https://www.desmos.com/calculator/i0n66pmccn
>
> We can also morph between a particular shape and its associated
> circle, where the amount of area enclosed remains constant as
> the morphing outline increases in length (as it deviates from the
> optimal circle shape):
>
> https://www.desmos.com/calculator/zgj65ahcdi
>
> Is there any math wizard here that has some ideas of how to
> obtain a similar morphing for the other shapes?
> What would be the most suitable way to accomplish this in
> desmos?
>

Last time I did a morph was when I interpolated between the points that
comprised both shapes. Iirc, I did it for the following:

https://youtu.be/8tnNzOVHPKU

https://youtu.be/dIIwmk3uRYA

On Tuesday, August 10, 2021 at 3:50:18 AM UTC+2, Chris M. Thomasson wrote:
> On 7/25/2021 5:17 PM, sobriquet wrote:
> >
> > In this geometric mandala, there are 7 constituent shapes and we can
> > use circles in corresponding colors to visualize the area of these
> > shapes (ordering them from the smallest at the top to larger ones
> > towards the bottom):
> >
> > https://www.desmos.com/calculator/2pnzxwil3y
> >
> > If we graph the length of the outline of each shape as a function
> > of the area it encloses, we can see how the shapes compare
> > to their associated circles in the way that they deviate from
> > optimal (in the sense of circles having the minimal length of outline
> > that is able to enclose a given amount of area):
> >
> > https://www.desmos.com/calculator/i0n66pmccn
> >
> > We can also morph between a particular shape and its associated
> > circle, where the amount of area enclosed remains constant as
> > the morphing outline increases in length (as it deviates from the
> > optimal circle shape):
> >
> > https://www.desmos.com/calculator/zgj65ahcdi
> >
> > Is there any math wizard here that has some ideas of how to
> > obtain a similar morphing for the other shapes?
> > What would be the most suitable way to accomplish this in
> > desmos?
> >
> Last time I did a morph was when I interpolated between the points that
> comprised both shapes. Iirc, I did it for the following:
>
> https://youtu.be/8tnNzOVHPKU
>
> https://youtu.be/dIIwmk3uRYA

Yeah, morphing itself is not so hard, but if you want to keep track of the area
remaining constant, that makes it more complicated.

On Tuesday, August 10, 2021 at 4:12:54 AM UTC-7, sobriquet wrote:
> On Tuesday, August 10, 2021 at 3:50:18 AM UTC+2, Chris M. Thomasson wrote:
> > On 7/25/2021 5:17 PM, sobriquet wrote:
> > >
> > > In this geometric mandala, there are 7 constituent shapes and we can
> > > use circles in corresponding colors to visualize the area of these
> > > shapes (ordering them from the smallest at the top to larger ones
> > > towards the bottom):
> > >
> > > https://www.desmos.com/calculator/2pnzxwil3y
> > >
> > > If we graph the length of the outline of each shape as a function
> > > of the area it encloses, we can see how the shapes compare
> > > to their associated circles in the way that they deviate from
> > > optimal (in the sense of circles having the minimal length of outline
> > > that is able to enclose a given amount of area):
> > >
> > > https://www.desmos.com/calculator/i0n66pmccn
> > >
> > > We can also morph between a particular shape and its associated
> > > circle, where the amount of area enclosed remains constant as
> > > the morphing outline increases in length (as it deviates from the
> > > optimal circle shape):
> > >
> > > https://www.desmos.com/calculator/zgj65ahcdi
> > >
> > > Is there any math wizard here that has some ideas of how to
> > > obtain a similar morphing for the other shapes?
> > > What would be the most suitable way to accomplish this in
> > > desmos?
> > >
> > Last time I did a morph was when I interpolated between the points that
> > comprised both shapes. Iirc, I did it for the following:
> >
> > https://youtu.be/8tnNzOVHPKU
> >
> > https://youtu.be/dIIwmk3uRYA
> Yeah, morphing itself is not so hard, but if you want to keep track of the area
> remaining constant, that makes it more complicated.

Trigonometry, defines sine and cosine, as after Pythagoras,
and after that there is for root 2 over 2, and so on.

A, tri-lateral-ometry, for, some, N-gonometry, is the idea that:
when an equilateral triangle is unhinged at one point, and,
the hinges each interpolate to pi radians, the point in motion
traces sine and cosine.

Instead of Pythagoras it's evolutive or as the tautochrone.
Unlike that the triangle as formed by the point at the origin
and the axes makes the Pythagoran, is for making, that
also the higher rank polygons, define a family of functions,
that, much like rational functions, makes for the orthogonal,
also what result that the same perimeter, and, area, results:
to reduce "shape with same area" to "shape with same perimeter".

There's also for means and the arithmetic and geometric,
or here basically for "interpolating between circles and squares
in terms of area and quadrature".