This Star Talk Cosmic Queries episode features hosts Neil deGrasse Tyson and Chuck Nice, joined by mathematician and YouTube educator Grant Sanderson (3Blue1Brown). The discussion delves into various facets of mathematics, including famous unsolved problems like the Riemann Hypothesis and the Navier-Stokes equations, as well as the intrinsically unsolvable problem of polynomial equations of degree five or higher, rooted in Galois theory. The conversation also explores commonly misunderstood concepts such as division by zero, complex numbers (rebranded as "clock numbers"), and the abstract nature of higher dimensions in mathematics. Grant Sanderson emphasizes the popular appeal of math despite its perceived intimidation, advocating for broader comfort with the subject and highlighting its role as a powerful language for understanding the universe.
Exploring the Universe Through Math: Unsolved Problems, Unknowable Answers, and Abstract Concepts
0:00 A Revolution in Math: The discussion opens by noting the 19th-century revolution in mathematics, stemming from the realization that polynomial equations of degree five or higher are fundamentally impossible to solve using standard algebraic formulas, a concept later attributed to Galois theory.
1:19 Introducing Grant Sanderson (3Blue1Brown): Neil deGrasse Tyson and Chuck Nice introduce their guest, Grant Sanderson, a mathematician and creator of the popular YouTube channel "3Blue1Brown" (7 million followers). Grant's mission is to make math more approachable and comfortable for the public.
2:56 Unsolved Math Problems: The hosts and Grant discuss famous unsolved mathematical problems, including the Clay Millennium Problems (each with a $1 million prize), which are often difficult to even articulate. They also mention "celebrity unsolved problems" that are easier to state, such as the Twin Prime Conjecture (are there infinitely many prime numbers two apart, e.g., 11 and 13?).
6:34 Math in Science (Navier-Stokes): An example of an unsolved problem with scientific origins is the Navier-Stokes equations, which describe fluid flow. Fundamental questions about their behavior, such as whether they predict infinite energy concentration, remain unanswered.
7:56 Intrinsically Unsolvable Problems (Quintic Equations): While quadratic, cubic, and quartic equations have solvable formulas, it was proven in the 19th century that no such formula exists for polynomial equations of degree five or higher using standard algebraic operations. This impossibility gave birth to modern abstract algebra.
11:06 The Legacy of Galois Theory: The proof of unsolvability for quintic equations is attributed to Évariste Galois, who developed Galois theory before his death in a duel at age 20. This work also proved other impossibilities, like trisecting an angle with a straightedge and compass.
14:00 Why Division by Zero is Undefined: Grant explains that division by zero is incoherent in practical terms (e.g., dividing one cupcake among zero people). However, in specific mathematical contexts like projective geometry, a "point at infinity" can be defined to conceptualize this.
16:24 Math and Cosmology (Singularities & Non-Euclidean Geometry): Neil deGrasse Tyson highlights singularity problems in cosmology (like black holes), where equations suggest infinite density—a mathematical "division by zero" indicating a limit to current physical theories, not the math itself. He points out how mathematical discoveries, like Riemann's non-Euclidean geometry in the 1800s, often lead scientific understanding (e.g., spacetime curvature).
19:09 Grant's Favorite Unsolved Problem (Riemann Hypothesis): Grant's personal favorite unsolved problem is the Riemann Hypothesis, one of the Millennium Problems. It concerns the distribution of prime numbers and connects them to the continuous Riemann Zeta function, painting a "beautiful picture" of prime number frequencies.
20:54 Understanding Complex Numbers: Grant addresses the "imaginary" label for complex numbers (square root of -1), calling it the "worst name ever." He clarifies that these numbers are natural for modeling cyclic processes, such as waves in quantum mechanics and electrical engineering. Their historical origin lies in the unexpected appearance of square roots of negatives when solving cubic equations, even when the final answer was real.
25:17 Circle Inversion in Conformal Geometry: Grant simplifies "circle inversion," a concept in conformal geometry, as treating a circle like a mirror to transform complex geometry problems into simpler ones by reflecting points from inside to outside and vice versa.
27:48 The Origin of Tensors: Tensors are introduced as higher-dimensional grids of numbers (beyond vectors as lists and matrices as 2D grids), used in physics (general relativity, spacetime curvature) and machine learning (representing words in high-dimensional spaces). They emerge when data naturally organizes in such complex structures.
30:43 New Paradigms in Modern Math: Grant asserts that new revolutionary mathematical paradigms are being discovered today, citing fields like algebraic geometry (unifying disparate parts of math) and category theory (a new "language" for mathematicians). He notes these are specialized tools for research and not intended for K-12 education.
34:50 The Irrationality of Pi and Proof by Contradiction: Gina's question about Pi's irrationality leads to Grant explaining that Pi is the irrational ratio between a circle's circumference and diameter. He demonstrates proof by contradiction using the irrationality of the square root of 2, a common mathematical technique to prove impossibility.
40:02 "Flatland" and Higher Dimensions: The conversation touches on Edwin A. Abbott's "Flatland," a book imagining a 2D world to illustrate the difficulty of comprehending higher dimensions. They then discuss what shapes they would be (Grant: circle, Neil: hexagon for aristocratic and tiling properties).
43:01 Periodic vs. Non-Periodic Tiling: They distinguish between periodic (repeating) and non-periodic (never repeating) tiling patterns, mentioning the recent discovery of a single tile ("the hat") that tiles non-periodically, and Penrose tiles (two rhombi) that tile non-periodically.
44:45 Mathematical Higher Dimensions: Grant clarifies that mathematicians' "higher dimensions" are not necessarily physical realities but abstract spaces where lists of numbers can be visualized as points, aiding in the conceptualization of complex data (e.g., in machine learning). The Klein bottle (a 4D object represented in 3D) and Mobius strip are examples of non-orientable shapes.
49:40 The Three-Body Problem and Chaos Theory: The three-body problem (predicting the motion of three interacting bodies) is discussed. Unlike Newton's two-body problem (which required inventing calculus), the three-body problem is intrinsically chaotic. Its "solution" is that it is unknowable and unpredictable without infinite precision, due to small initial errors blowing up quickly.
53:27 The Power of Mathematics: Neil deGrasse Tyson concludes by emphasizing math's role as a powerful model of reality, allowing scientists to "peer through doorways" and explore the universe without leaving their chairs. Math's predictions, when true, bolster confidence in the unity of math and the universe.
54:51 3Blue1Brown Promotion: Grant Sanderson reiterates his YouTube channel name (Three Blue, One Brown / 3B1B) and social media handles.