Summary of How to lie using visual proofs

00:00:00 - 00:15:00

This video discusses how to lie using visual proofs, and how to be careful when applying limiting arguments. It shows how the limit of the lengths of curves can be different from the length of the limits of the curves, and why this is a good counter example to show why π is not equal to 4.

• 00:00:00 The first proof shown in the video is for the surface area of a sphere, and it starts with dividing the sphere into vertical slices. Symmetrically, all of the slices from the southern hemisphere are unraveled and symmetrically, all of the slices from the northern hemisphere are unraveled. The base of the shape came from the circumference of the sphere, and its length is 2π times the radius of the sphere. The other side of the shape came from the height of one of the wedges, and its length is π/2 times the radius of the sphere. The proof is elegant and has a surprising outcome- the true surface area of a sphere is 4π R squared. The second proof is for a simple argument for the fact that π is equal to four. It starts with a circle and asks how can we figure out its circumference. After all π is by definition the ratio of this circumference to the diameter of the circle, we start off by drawing the square whose side lengths are all tangent to that circle. The perimeter of the square is 8. Then, and some of you may have seen this before, the argument proceeds by producing a sequence of curves, all of whom also have this perimeter of 8, but which
• 00:05:00 In this video, an Euclid-style proof is given for the claim that all triangles are isosceles. The proof starts by drawing the perpendicular bisector for the line BC and labeling the intersection point D. Next, the angle bisector at A is drawn and labeled alpha. The point where these two intersect is P is then drawn and labeled. Next, new lines are drawn to figure out what things must be equal and some conclusions are drawn. For instance, the line from P which is perpendicular to the side length AC is drawn and labeled E, and the line from P down to the other side length AB is drawn and labeled F. The first claim is that AFP is the same as AEP, which follows from symmetry across the angle bisector. Next, the triangle with corresponding two side lengths over here is congruent to its reflection across the perpendicular bisector and is CPD. The last claim is that the side FB is the same as the side EC, which is derived from adding up the two equations AF+FB and AE+EC. So all triangles really are equilateral, and the proof leaves us with three possible explanations for why that might be the case.
• 00:10:00 The author discusses two visual proofs, one for the area of a circle and one for the width of a wedge on a sphere. The first proof works well when the geometry is preserved, but the second proof fails because the geometry is not preserved when the pieces are rearranged.
• 00:15:00 The video discusses how to lie using visual proofs, and how to be careful when applying limiting arguments. It shows how the limit of the lengths of curves can be different from the length of the limits of the curves, and why this is a good counter example to show why π is not equal to 4.