From: 3blue1brown
Mathematics often leverages the visual nature of geometry to understand abstract concepts, but this approach has inherent limitations, particularly when dealing with higher dimensions [00:00:03].
The Appeal of Geometric Reasoning
There is an inherent beauty in reasoning geometrically in two and three dimensions [00:00:06]. This method offers a strong back and forth between numerical pairs or triplets and spatial understanding, which the human visual cortex is adept at processing [00:00:10].
For instance, visualizing a circle with radius 1 centered at the origin allows for the conceptualization of every pair of numbers (x,y) satisfying the numerical property x² + y² = 1 [00:00:19]. Many facts that appear obscure in a purely analytic context become clear geometrically, and vice versa [00:00:35]. This back and forth between sums of squares and circles/spheres is fundamental [00:01:02]. Examples include:
- Connecting Pi to number theory and primes [00:01:07].
- Visualizing all possible Pythagorean triples [00:01:11].
- Using topological facts about spheres to solve counting puzzles, as in the Borsuk-Ulam theorem [00:01:16].
The ability to frame analytic facts geometrically is extremely useful for mathematics [00:01:25].
Limitations: The “Tease” of Higher Dimensions
The utility of geometric intuition becomes a “tease” when mathematicians venture into higher dimensions, dealing with quadruplets, quintuplets, or even 100-tuples of numbers [00:01:31]. The constraints of our physical space limit our intuitions about geometry, causing us to lose the valuable back and forth between numbers and spatial understanding [00:01:40]. It is challenging to conceptualize lists of many numbers as individual points in some space [00:01:52].
For mathematicians, computer scientists, and physicists, problems involving lists of more than three numbers are common [00:02:02]. The standard approach to doing math in higher dimensions is to use two and three dimensions for analogy, but to fundamentally reason analytically [00:02:10]. This is analogous to a pilot relying primarily on instruments rather than sight when flying through clouds [00:02:19].
A Hybrid Approach: Visualizing Analytic Reasoning
To bridge the gap between purely geometric and purely analytic views, a hybrid method can be employed to make analytic reasoning more visual, even in arbitrarily high dimensions [00:02:28]. This approach aims to make counterintuitive higher-dimensional phenomena more intuitive [00:02:55].
The focus is on higher-dimensional spheres. For example, a four-dimensional sphere with radius one centered at the origin is defined as the set of all quadruplets of numbers (x,y,z,w) where the sum of their squares is one (x² + y² + z² + w² = 1) [00:03:02]. Direct projections of 4D spheres into 3D can be confusing and don’t reflect the true nature of doing math with such objects [00:03:20].
The “Sliders” and “Real Estate” Analogy
A concrete way to think about higher dimensions is to be literal: visualize four separate numbers for 4D space [00:03:49].
- Sliders: Picture multiple vertical number lines, each with a slider representing a coordinate (x, y, z, w, etc.) [00:03:55]. Each configuration of these sliders represents a point in N-dimensional space (e.g., a quadruplet of numbers) [00:04:00].
- Unit Sphere Condition: For a point to be on a unit sphere centered at the origin, the sum of the squares of these slider values must equal one [00:04:04]. Understanding movements on the sphere means understanding which slider movements satisfy this condition [00:04:16].
- “Real Estate” Analogy: The term “real estate” is used to describe the value of a coordinate squared (e.g., x² is x’s “real estate”) [00:04:38]. The total “real estate” (sum of squares) must always be one for a unit sphere [00:04:46]. Moving around the sphere corresponds to a constant exchange of “real estate” between variables [00:04:51].
- Cost of Real Estate: “Real estate” is “cheap” near zero (a small change in the slider corresponds to a large change in square value) and “expensive” away from zero (a large change in the slider corresponds to a small change in square value) [00:05:03]. This leads to a “piston looking dance motion” where sliders move more slowly away from zero [00:06:17].
- The square root of the total real estate among all coordinates gives the distance from the origin [00:06:43].
This “bug on the surface” perspective, focusing on local movements and real estate exchange, provides a foothold for thinking about high-dimensional shapes more concretely [00:08:02] [00:23:36].
A Classic Example: The Inner Sphere Radius in N Dimensions
Consider a box of side length 2 centered at the origin (e.g., in 2D, a 2x2 square with corners at (±1, ±1)) [00:09:10]. Place unit spheres (radius 1) at each corner [00:09:21]. The problem is to find the radius of the sphere centered at the origin that is just large enough to be tangent to these corner spheres [00:09:30].
2 Dimensions
- Setup: A 2x2 square with unit circles at its four corners [00:09:10].
- Calculation: The distance from the origin to a corner (e.g., (1,1)) is √(1² + 1²) = √2 ≈ 1.414 [00:09:48]. Subtracting the radius of a corner circle (1), the inner circle’s radius is √2 - 1 ≈ 0.414 [00:10:02]. This seems reasonable [00:10:11].
- Slider View: For a corner circle centered at (1,-1), its “real estate” is defined by (x-1)² + (y-(-1))² = 1² [00:12:43]. The point of tangency with the inner circle occurs when x and y coordinates are the same [00:13:20], specifically when the “real estate” is shared evenly [00:13:31]. At this point, x and y are both less than 0.5, so the total x² + y² (relative to the origin) is less than 0.5² + 0.5² = 0.5 [00:14:55].
3 Dimensions
- Setup: A 2x2x2 cube with eight unit spheres at its corners [00:15:17].
- Calculation: The distance from the origin to a corner (e.g., (1,1,1)) is √(1² + 1² + 1²) = √3 ≈ 1.73 [00:11:48] [00:11:23]. Subtracting the radius (1), the inner sphere’s radius is √3 - 1 ≈ 0.73 [00:11:29]. This also seems reasonable [00:11:43].
- Slider View: For a corner sphere at (1,1,1), the point closest to the origin (tangent point) is where x, y, and z are equal [00:15:22]. They must be slightly beyond 0.5 because 0.5² = 0.25, and with three coordinates, 3 * 0.25 = 0.75, which is not enough for the total unit of real estate shared among the corner sphere’s dimensions [00:15:43]. Relative to the origin, this point has x² + y² + z² less than 0.75, confirming the inner sphere is smaller than the corner spheres [00:16:03].
Higher Dimensions: The Surprising Twist
Something unexpected happens to the inner radius as dimensions increase [00:11:48]. The ability to use sliders helps to understand this [00:12:04].
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4 Dimensions:
- Setup: A 2x2x2x2 hypercube with 16 unit spheres at its corners [00:16:37].
- Calculation: The distance from the origin to a corner (e.g., (1,1,1,1)) is √(1² + 1² + 1² + 1²) = √4 = 2 [00:17:55]. Subtracting the radius (1), the inner sphere’s radius is 2 - 1 = 1 [00:18:03].
- Slider View: The point on a corner sphere (e.g., centered at (1,1,1,1)) closest to the origin is when all four coordinates are equally splitting the corner sphere’s unit of real estate [00:17:04]. This happens precisely when each coordinate is 0.5 [00:17:09]. When viewing this as a point on the inner sphere relative to the origin (0,0,0,0), each of these four coordinates still has 0.5² = 0.25 units of real estate. The total is 4 * 0.25 = 1 [00:17:34].
- Result: The inner sphere has precisely the same size (radius 1) as the corner spheres [00:17:50]. This means if all real estate goes to one coordinate, that coordinate can reach 1 (e.g., point (1,0,0,0)), touching the hypercube’s face [00:18:20]. This is a purely four-dimensional phenomenon and cannot be intuitively crammed into smaller dimensions [00:18:45].
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5 Dimensions:
- Setup: A 2x2x2x2x2 hypercube with 32 unit spheres at its corners [00:18:57].
- Calculation: The distance from the origin to a corner (e.g., (1,1,1,1,1)) is √5 ≈ 2.236 [00:19:50]. The inner sphere’s radius is √5 - 1 ≈ 1.24 [00:19:50].
- Slider View: The point on a corner sphere closest to the origin has all five coordinates equally splitting the real estate. Each coordinate will be slightly higher than 0.5 (as 5 * 0.25 = 1.25, too much for 1 unit) [00:19:06]. When viewed from the origin, this configuration has much more than one unit of real estate [00:19:27].
- Result: The inner sphere is larger than the corner spheres, and its radius (1.24) means it actually “pokes outside” the 2x2x2x2x2 box [00:20:12] [00:20:05].
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10 Dimensions:
- Calculation: The inner sphere’s radius is √10 - 1 ≈ 2.16 [00:20:58].
- Slider View: The point on a corner sphere closest to the origin has all ten coordinates splitting the real estate evenly [00:20:39]. This appears very far from the origin, indicating a large amount of real estate for the inner sphere [00:20:47].
- Result: The inner sphere is more than twice as large as the corner spheres [00:21:13]. It even “pokes outside” of a 4x4x…x4 outer bounding box that would enclose all corner spheres [00:21:36]. As dimensions increase, the inner sphere continues to grow without bound, and the proportion of the inner sphere lying inside the box decreases exponentially towards zero [00:22:22]. This happens because the face of the box is always 2 units from the origin (moving along one axis), while the corner’s distance from the origin is √N (where N is the number of dimensions), which determines the inner sphere’s radius [00:21:50] [00:22:05].
Conclusion
The slider method, while a direct representation of analytic descriptions, re-frames these instruments to better utilize the brain’s image processing capabilities [00:24:20]. It allows for a more concrete and less metaphysical understanding of higher dimensions, enabling mathematicians to inquire about other properties of high-dimensional shapes [00:22:41]. Just because something cannot be fully visualized, it does not mean it cannot be thought about visually [00:24:42].