An n-ball Between n-balls
The article explores a geometric thought experiment illustrating high-dimensional spaces, focusing on the behavior and volume relationships of n-balls and n-cubes as dimensions increase, revealing counterintuitive results.
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The article discusses a geometric thought experiment that illustrates the complexities of high-dimensional spaces through an interactive visual model. It begins with a 4x4 square containing four blue circles at each corner and a central red circle, which is maximized without overlapping the blue circles. As the model transitions to three dimensions, the circles become spheres, and the red sphere grows larger while new spheres are introduced. The article defines an n-dimensional construct as an n-cube with n-balls positioned between its vertices and the center. It explores the behavior of these shapes as dimensions increase, noting that the red ball can extend beyond the confines of the enclosing box in higher dimensions, a counterintuitive result. The article also discusses the volume relationships between n-balls and n-cubes, highlighting that while the volume of the n-ball decreases as dimensions increase, the surrounding space grows, leading to the perception of the n-ball being "spiky." The mathematical implications of these constructs are examined, including volume calculations for various dimensions, revealing that at certain dimensions, the volume of the n-ball can exceed that of the enclosing box significantly.
- The thought experiment illustrates the counterintuitive nature of high-dimensional geometry.
- The model transitions from 2D to 3D, showing how shapes evolve with added dimensions.
- In higher dimensions, the central n-ball can extend beyond the enclosing n-cube.
- Volume relationships between n-balls and n-cubes change as dimensions increase.
- The article emphasizes the mathematical complexities and visualizations of high-dimensional constructs.
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- There is confusion and debate over the distinction between "ball" and "sphere" in mathematical terminology.
- Many commenters express fascination with the concept of the curse of dimensionality and its implications.
- Visual aids and animations are highly appreciated, with several users noting their impact on understanding the concepts.
- Some users reference additional resources, such as videos and previous articles, to further explore the topic.
- There is a sense of wonder and challenge in grasping the complexities of high-dimensional geometry.
15 comments[0]: https://en.m.wikipedia.org/wiki/Curse_of_dimensionality
I don’t know that there would even be anything interesting to say about that.
That article doesn't have the nice animations, but it is from 14 years ago ...
https://news.ycombinator.com/item?id=12998899
https://news.ycombinator.com/item?id=3995615
And from October 29, 2010:
“…At what dimension would the red ball extend outside the box?”
If anyone has o1-preview it’d be interesting to hear how well it does on this.
We don't have special words for the voluminous versions of other 3D shapes, so why do spheres need one?
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The Banach–Tarski paradox challenges geometric intuition by dividing a ball into subsets that can form two identical copies without changing volume. Axiom of choice and group actions play key roles.
The Point of the Banach Tarski Theorem
The Banach-Tarski theorem challenges common sense by showing a solid ball can be split into pieces to form two identical balls in 3D space. It questions measurement principles and the Axiom of Choice's role in resolving mathematical paradoxes.
Spikey Spheres (2010)
The text explores the challenges of visualizing high-dimensional spheres, emphasizing the complexity of navigating such spaces and the limitations of three-dimensional intuition in understanding their properties.
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