[This was an afterthought to the kids, but you may like the Cartesian "chord blood" and the line intersector.] [Just kids]: (Aside: This dissection is of a very restricted type in that the pieces permute without rotating.) This should work for rectangles other than Golden. Which ones? This is what I meant when I sent the old construction and asked for the applicable range of aspect ratios. (Nobody answered.) Instead of Golden, can the rectangle be tall and skinny? Extremely squat? Same answer for both constructions? Numbers, please. Younger guys: Note that, if the rectangle is a⨉b, the radius of the 3rd arc is √a√b, the geometric mean. (Old name: mean proportional.) Fairly amazing geometry thm: When two chords intersect, the product of the "halves" of one equals the product of the "halves" of the other. Cartesian proof with four arbitrary points on the unit circle, In[599]:= Assuming[{a, b, c, d} \[Element] Reals, FullSimplify@ComplexExpand[Abs[I^a - #] Abs[# - I^b] == Abs[I^c - #] Abs[# - I^d] &@ Echo@Interseg[I^a, I^b, I^c, I^d]]] ((I^c-I^d) Im[I^b Conjugate[I^a]]+(I^a-I^b) Im[I^c Conjugate[I^d]])/Im[(I^c-I^d) (Conjugate[I^a]-Conjugate[I^b])] Out[599]= True (This takes "forever" w/o the ComplexExpand.) Note the "Echo"ed previous line is a fully general line intersector, if you replace I^a by z1, ..., I^d by z4, and z1,z2 determine the 1st line and z3, z4 determine the 2nd. Mma must offer this functionality somewhere. ? Note also: The "chords" can intersect outside the circle, in which case they are secants! —Bill On Sun, Jun 9, 2019 at 4:45 AM Bill Gosper <billgosper@gmail.com> wrote:
On Tue, May 21, 2019 at 6:17 PM Bill Gosper <billgosper@gmail.com> wrote:
Golden rectangle↔︎square <http://gosper.org/gold2square.gif> I forget where I stole this, but can you find the superfluous step I just noticed? (Not the retrace, which was to tell you a dimension.) —rwg
SPOILER: The arc centered on the top edge of the rectangle is superfluous.
(Rebuild in progress)
Nicer golden rectangle to square <http://gosper.org/gold2square2.gif> I can re-export this to loop if people want.
I think it loops if you download and view it with Finder.
—rwg