The things you'll find in higher dimensions
Zach Star・24 minutes read
Connecting points to form shapes like quadrilaterals and triangles results in parallel lines and segments, leading to insights on polyhedra, Euler's formula, higher dimensions, unique shapes like the hyper diamond, sphere volumes, and optimal sphere packing for error correction in communication channels. As dimensions increase, the volume of spheres changes, with different optimal packing methods and ratios between spheres and hypercubes in different dimensions.
Insights
- Connecting midpoints of two sides of a triangle creates parallel line segments, forming a new triangle.
- The relationship between the volume of a hyper sphere and a hypercube shifts in higher dimensions, with the hyper sphere eventually exceeding the hypercube's volume after 262 dimensions.
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Recent questions
What shapes result from connecting four dots on a paper?
Parallelogram
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Summary
00:00
Geometry and Dimensions: Exploring Mathematical Relationships
- Connecting four dots on a paper to form a quadrilateral always results in a parallelogram.
- The theorem states that connecting midpoints of two sides of a triangle yields a line segment parallel to the third side.
- By connecting midpoints of two sides of a triangle and forming another triangle, parallel line segments are created.
- Exploring three-dimensional polyhedra, only five Platonic solids meet specific criteria.
- Euler's characteristic formula (vertices - edges + faces = 2) applies to convex polyhedra and even a sphere.
- The Klein bottle, existing in four dimensions, is a shape that cannot self-intersect in three dimensions.
- Knot theory delves into unknotting numbers, showcasing how knots can be manipulated in different dimensions.
- Higher dimensions introduce complexities like four-dimensional objects moving through three-dimensional space.
- Four-dimensional dice, or platonic solids, have counterparts in the fourth dimension, with unique shapes like the hyper diamond.
- Kissing numbers, determining the maximum number of spheres touching a central sphere, vary in different dimensions, with some unknown values.
13:19
"Sphere Packing in Higher Dimensions Explained"
- A four-dimensional sphere with a radius of one has a volume of approximately 4.935, while a five-dimensional unit sphere has a volume of 5.264.
- As dimensions increase, the volume of a sphere with a radius of one initially increases but then decreases after the fifth dimension.
- The volume of a unit sphere decreases as dimensions increase due to the ratio of volumes changing with additional dimensions.
- Sphere packing involves arranging identical spheres to fill space optimally, with the best arrangement taking up about 74% of the space for spheres.
- In higher dimensions, gaps between spheres increase, but in the eighth dimension, new gaps allow for an optimal arrangement taking up about 25% of the space.
- Sphere packing has applications in coding theory for error correction in communication channels.
- In higher dimensions, the relationship between the volume of a hyper sphere and a hypercube changes, with the hyper sphere eventually surpassing the hypercube in volume after 262 dimensions.
- The optimal packing method for arranging spheres changes with dimensions, with aligning centers on the same line being best until around 50,000 to 100,000 four-dimensional spheres, after which clumping becomes more efficient.
