1. Planimeters and Isoperimetric Inequalities Robert L. Foote Wabash College

2. Isoperimetric Problem Given a fixed length for the boundary of a region in the plane, what shape has the largest area? Answer: A circle, attributed to Dido, Queen of Carthage, Greek and Roman mythology, appears in Virgil’s Aeneid, ca. 25 BC Dido’s problem. To enclose the largest possible area by the sea using an ox hide. Solution: Cut hide into thin strips and form a semi-circle. Not proven until early 1900s! How are area and perimeter related for a circle?

3. Isoperimetric Inequality For every region, Furthermore, if and only if the boundary is a circle. First proved by Hurwitz (1902) using Fourier series. The inequality Is “sharp.” The inequality implies that circles are solutions The sharp part implies that circles are the only solutions This solves the isoperimetric problem

4. Polar and Linear Planimeters Jacob Amsler, 1854 Polar Planimeter The wheel rolls and slides – it measures the component of its motion perpendicular to the tracer arm. The area is proportional to the net roll of the wheel (not obvious!) Linear Planimeter

5. Line segment sweeping out a signed area Positive Negative

6. Component of dm in direction of N A Formula for Signed Area p qm N dm Moving segment has four degrees of freedom Motion of midpoint (2 degrees) Rotation about midpoint Change length Roll (signed distance) of a wheel at m Planimeter: Moving segment sweeping out area with a wheel attached p qm N dm dσmdσm

7. The Area Difference Theorem If the endpoints of a moving segment each go around a region CCW, the signed area swept out by the segment is Intuitive reason

8. Recall (if you’ve seen Green’s Theorem) Proof of the Area Difference Theorem … To show: the signed area here is

9. Proof of the Area Difference Theorem … p q m N dm dp dq Integrates to Integrates to 0

10. Planimeter: A moving segment of fixed length with a wheel attached Location of wheel Roll of wheel determined by λ

11. Now three degrees of freedom Translation forward/backward, measured by dσ λ Rotation about wheel, measured by dθ Translation sideways, doesn’t contribute to dA Consider No rotation: Rotation about wheel:

12. How a Planimeter Works! Integrate to get area: Ω Area Difference Theorem: Right endpoint goes around ∂Ω: A R = A Ω Left endpoint goes around no area: A L = 0 No net rotation of segment: Area is proportional to roll of wheel Net roll of wheel doesn’t depend on location!

13. The Prytz “hatchet” Planimeter 1875, Holger Prytz, Danish mathematician and cavalry officer Behaves like a bicycle Front wheel: tracer point Rear wheel: chisel edge Economical alternative to Amsler’s planimeters

14. It can measure area! σ is the net disp. of the chisel (red arc) Error

15. To Prove: Isoperimetric Inequality if and only if the boundary is a circle We’ll do better: Isoperimetric Defect

16. Put wheel at right endpoint (λ = 1) so it rolls along ∂Ω. New: Have planimeter make one CCW rotation. This and size of region put geometric constraints on ℓ. As before, A R = A Ω and A L = 0. Trace differently … ℓ = half-width ℓ = circumradius ℓ = inradius

17. Goal: Currently have Really, just complete the square in ℓ and rearrange!

18. q N dq Key Observation is a component of ds

19. This implies Need a more geometrically meaningful expression Solve for and substitute … When do we have ?

20. Suppose Then So ∂Ω is the circle. But Ω is contained in the circle. The radius of the circumscribing circle determines ℓ. Ω “Sharp” part of the Isoperimetric Inequality

21. if and only if the boundary is a circle B has geometric significance B = 0 iff the boundary is a circle Bonnesen found several of these, 1920's Osserman, Amer. Math. Monthly, Jan 1979 Bonnesen type of Isoperimetric Inequality

22. Isoperimetric Inequality in Spherical and Hyperbolic Geometries K is the Gaussian curvature of the geometry; for a sphere. Get equality if and only if the region is circular.

23. p q p q Segments sweep out area differently than in Euclidean geometry … … but similar proofs work for planimeters and the isoperimetric inequality.

24. Amsler's Spherical Polar Planimeter, 1884 Never manufactured. One prototype built. Many other types of isoperimetric inequalities: higher dimensions, other geometries, in physics …

25. A Plethora of Planimeters! A Plethora of Planimeters! What’s for sale on eBay? What’s for sale on eBay? Thanks! Check out the planimeters on display

27. Construction of tracks