This tie is a favorite of mine and was actually a Christmas gift from a former student a few years ago. But I never really thought about it in any mathematical way until recently when a current student mentioned that it was “like those SAT problems with the shaded areas.”
So I wondered, “what proportion of the tie’s area is taken up by all those circular segments?”
I snapped a closer picture of the tie’s pattern and inserted it into GeoGebra. I figured a good place to start would be to construct a circle using one of the arcs, a method I learned simultaneously from one of my colleagues and this video from James Tanton.
First, I draw a chord of the arc and construct the perpendicular bisector of that chord:
Next, I pick a different point on the arc and repeat the chord-perpendicular-bisector construction.
The intersection of the two perpendicular bisectors (point D) is the center of a circle containing the points A, B, and C!
From here, I can ask GeoGebra to do a few measurements in its own coordinate system. The length of a side of the square (also the length of my first chord) is 3.22; the radius of the circle is 2.74. The central angle subtended by the arc measures 72.14°.
The area of the circular segment is then:
And the proportion p that all the segments take up on the tie is:
About 44%!
A square is inscribed into a unit circle cuts of the circle 4 segments of total area π – 2. This is also the total area of the segments inside the square obtained by the reflections in the sides. The proportion the latter occupy in the square is [2-((π-2)]/2 ≈0.4292…
Aren’t you assuming that the square is inscribed in a circle? My construction seems to show that the figure encompassing a single square is not a circle. I’m intrigued by how close my approximation is to your method though.
Yes, of course, this is what I assumed. I would vouch that the shape is a circle. The construction is rather approximate.
Now that I think of it, it might be safer to not make this assumption that the arcs are those of a circumscribed circles. This weighs against the unavoidable inaccuracy in picking a third point an arc. The winner? I’ll concede that vouching for a specific case was an immature act, while making use of GeoGebra served an excellent example of its usefulness and power.