# How to lie using visual proofs

TLDRThis video script explores the deceptive nature of visual proofs in mathematics, presenting three examples of incorrect proofs and explaining their fallacies. The first proof incorrectly calculates the surface area of a sphere, while the second attempts to prove that pi equals 4 using a sequence of approximating curves. The third example, a Euclidean-style proof, claims all triangles are isosceles, highlighting the importance of rigor and critical thinking to avoid hidden assumptions and errors. The script emphasizes that while visual intuition can aid understanding, it cannot replace the need for mathematical precision.

### Takeaways

- 🔍 The video discusses three visual proofs that are mathematically incorrect but appear convincing, highlighting the importance of rigor in mathematical proofs.
- 🌐 The first proof incorrectly calculates the surface area of a sphere by unfolding it into a rectangle shape, leading to the wrong formula π²r² instead of the correct 4πr².
- 📏 The second proof attempts to show that pi equals 4 by approximating a circle with a square and then a sequence of curves, which is flawed due to the jagged nature of the approximations.
- 📐 The third proof, which claims all triangles are isosceles, uses Euclidean geometry but contains a subtle error in the assumption about the location of the intersection point of the angle bisector and perpendicular bisector.
- 🧩 The video emphasizes that visual intuition can be misleading and that critical thinking is necessary to identify hidden assumptions and edge cases in mathematical proofs.
- 📚 It is crucial to differentiate between the limit of the lengths of curves and the length of the limit of curves, as they are not necessarily the same.
- 🔍 The sphere example shows that the geometry of a curved surface cannot be accurately represented by flattening it into a flat shape without losing geometric information.
- 📉 The video uses a rearrangement puzzle to illustrate how area can appear or disappear due to the non-linear nature of curved surfaces, cautioning against assuming straight lines without verification.
- 📈 The concept of Gaussian curvature is introduced as a fundamental difference between the geometry of flat space and curved surfaces, such as a sphere.
- 📝 The video concludes by stressing that while visual proofs can aid understanding, they cannot replace the need for rigorous mathematical proof and critical analysis.
- 🤓 It challenges viewers to identify the subtle errors in the presented proofs, emphasizing the need for attention to detail and a deep understanding of mathematical principles.

### Q & A

### What is the first fake proof discussed in the script about?

-The first fake proof is about a formula for the surface area of a sphere. It involves subdividing the sphere into vertical slices and unraveling them to form an approximate rectangle, leading to the incorrect formula of pi squared times r squared for the surface area instead of the correct 4 pi r squared.

### Why is the first proof for the surface area of a sphere incorrect?

-The first proof is incorrect because it assumes that as the sphere is subdivided into finer slices, the shape formed approaches a perfect rectangle. However, the edges of the slices do not form straight lines but rather bulge outward due to the curvature of the sphere, leading to an overlap that is not accounted for in the proof.

### What is the argument presented for pi being equal to 4?

-The argument for pi being equal to 4 starts with a circle of radius 1 and a square inscribed within it. By creating a sequence of curves that approximate the circle and have the same perimeter of 8, the argument suggests that the limit of these curves is the circle, and therefore, pi would be equal to 4, which is incorrect.

### How does the script explain the limit of a sequence of curves?

-The script explains the limit of a sequence of curves by considering a specific value of the parameter t and evaluating the sequence of functions at this point. As the number of iterations approaches infinity, the limit is a well-defined point on the circle, leading to the definition of a new function c infinity, which represents the circle itself.

### What is the key issue with the argument that pi equals 4 based on the limit of jagged curves?

-The key issue is that the limit of the lengths of the curves is assumed to be the same as the length of the limit of the curves. However, this is not necessarily true, as the example shows that the limiting curve is a circle with a perimeter of 8, but the actual circumference of a circle is 2 pi times the radius, not 8.

### What is the claim made in the Euclid-style proof about triangles?

-The claim made in the Euclid-style proof is that all triangles are isosceles, meaning that any two sides of a triangle are necessarily equal in length. This is proven by drawing specific lines and using congruence relations, but the proof is flawed.

### What is the flaw in the Euclid-style proof that all triangles are isosceles?

-The flaw in the proof is the assumption that the angle bisector intersects the side of the triangle at a point that is between the two vertices. In reality, for many triangles, the intersection point sits outside the triangle, which invalidates the subsequent congruence claims and the conclusion that all triangles are isosceles.

### Why is it important to be cautious with visual proofs and arguments?

-Visual proofs and arguments are important for providing intuition, but they can be misleading if they rely on hidden assumptions or overlook edge cases. Critical thinking and rigorous mathematical proof are necessary to ensure that the conclusions drawn are correct.

### What is the difference between the geometry of a curved surface and flat space?

-The geometry of a curved surface, such as a sphere, has Gaussian curvature, which means that you cannot flatten it out without losing geometric information. In contrast, flat space does not have this curvature, and shapes can be transformed without losing their properties.

### How does the script use the concept of limits to discuss the approximation of areas under curves in calculus?

-The script discusses the use of limits in calculus to approximate areas under curves by using rectangles. As the rectangles become thinner with finer subdivisions, the sum of their areas approaches the actual area under the curve. However, it is crucial to be explicit about the error between the approximations and the true area to ensure rigor.

### Outlines

### 💡 Misleading Geometric Proofs

This paragraph discusses the pitfalls of three flawed geometric proofs. The first proof incorrectly calculates the surface area of a sphere by subdividing it into vertical slices and attempting to form a rectangle, leading to an erroneous formula. The second proof, which claims that pi equals 4, uses a square inscribed in a circle and a sequence of approximating curves, but fails to account for the jaggedness of the approximations. The paragraph emphasizes the importance of rigor in mathematical proofs and the limitations of visual intuition, highlighting the need for a more precise mathematical approach.

### 🔍 The Importance of Rigor in Mathematical Proofs

The second paragraph delves into the necessity of mathematical rigor, using the example of a proof that all triangles are isosceles to illustrate common mistakes. It points out that even with seemingly valid congruence relations, the proof fails due to an incorrect assumption about the location of a constructed point. The paragraph also contrasts the incorrect sphere proof with a valid proof involving pizza slices, emphasizing the importance of considering the geometry of curved surfaces and Gaussian curvature. It concludes by cautioning against the assumption that the limit of lengths of curves equates to the length of the limit of curves.

### 📏 The Nuances of Limiting Arguments in Calculus

This paragraph examines the subtleties of limiting arguments, particularly in calculus, using the example of approximating the area under a curve with rectangles. It stresses the importance of being explicit about the error between approximations and the true area, and the need to ensure that the cumulative area of the error approaches zero as the approximation becomes finer. The paragraph also revisits the incorrect proof that pi equals 4, explaining why the limit of the lengths of the approximating curves does not equal the length of the limit of the curves, thus serving as a counterexample to a common misconception.

### 🧐 The Subtleties of Visual Intuition in Mathematical Proofs

The final paragraph emphasizes the limitations of visual intuition in mathematical proofs. It revisits the incorrect proof that all triangles are isosceles, pointing out the subtle error in the assumption about the location of a constructed point. The paragraph argues that while visual proofs can be illustrative, they cannot replace the need for critical thinking and attention to detail. It concludes by stressing the importance of identifying hidden assumptions and edge cases in mathematical reasoning.

### Mindmap

### Keywords

### 💡Surface Area

### 💡Vertical Slices

### 💡Limit

### 💡Pi (π)

### 💡Tangent

### 💡Approximation

### 💡Euclidean Proof

### 💡Congruence

### 💡Angle Bisector

### 💡Hidden Assumptions

### Highlights

The video presents three fake proofs to illustrate subtle mathematical concepts.

The first fake proof discusses a formula for the surface area of a sphere using vertical slices.

The sphere is subdivided and unraveled to form an approximate rectangle for area calculation.

The area of the sphere is incorrectly concluded to be pi squared times r squared.

The true surface area of a sphere is 4 pi r squared, contrasting the incorrect proof.

A second proof argues that pi equals 4 by approximating a circle with a square and jagged curves.

The perimeter of the square and the curves is maintained at 8, leading to the incorrect conclusion.

The importance of rigor and proofs in mathematics is emphasized over visual intuition.

The third proof attempts to show that all triangles are isosceles using Euclidean geometry.

A triangle's sides are assumed to be equal through a series of geometric constructions.

The proof concludes incorrectly that any triangle is equilateral due to a hidden assumption.

The video challenges viewers to identify the flaw in the isosceles triangle proof.

The difference between valid and invalid visual proofs in geometry is explored.

The geometry of a curved surface like a sphere is fundamentally different from flat space.

Limiting arguments in calculus require careful consideration of errors and approximations.

The video concludes by emphasizing the necessity of critical thinking in mathematics.

Visual intuition can be misleading and should not replace rigorous mathematical proof.

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