Volumes in Higher Dimensions

(Image: Freepik)

A prior post of this blog, Atomic Interstitial Sizes in Higher Dimensions, explored how the size of gaps between tightly packed spheres changes as one moves from two-dimensional to three-dimensional and on to higher dimensions.

Grant Sanderson, host of the 3Blue1Brown YouTube channel (I highly recommend following this channel), gave an exceptional lecture explaining how to calculate the volume of high-dimensional spheres. That lecture is: https://www.youtube.com/watch?v=fsLh-NYhOoU&t=3467s. While he doesn't directly speak about the interstitial sizes of the gaps found in my post, he does approach the problem by looking at the ratio of the sphere to cube volume in higher dimensions. 

Summary of Exoplanets Discovered in 2025

An artist's illustration of the various exoplanets found, with rows of colorful planets of all colors and sizes over a dark background (Image credit: NASA's Goddard Space Flight Center)

Space.com wrote a summary of exoplanets discovered in 2025. The first exoplanet was discovered in 1995, and the number now exceeds 6000. With the James Webb Space Telescope (JWST) and other modern telescopes, this total is likely to grow by the thousands in the next few years.

One of the more interesting planets is 2M1510(AB)b which has an orbit perpendicular to the plane of the orbits of its binary stars: https://cdn.jwplayer.com/previews/WXbSE5GI.

Prior posts on JWST:

Number of planets in the universe:

https://jamesmacmath.blogspot.com/2022/01/number-of-planets-in-universe.html

Earliest star formation:

https://jamesmacmath.blogspot.com/2025/05/jwst-detects-earliest-formed-galaxies.html

Top ten JWST photographs:

https://jamesmacmath.blogspot.com/2024/07/top-ten-photographs-of-james-webb.html

Columbian Mathematicians

(Fedrico Ardila - Image: QuietSisyphus)

Colombia may be best known for coffee, emeralds, and magical realism, but it has quietly produced some extraordinary mathematicians. Federico Ardila (combinatorics rockstar, MIT PhD, mentor to dozens of underrepresented students), Antanas Mockus (the mayor who used game theory to transform Bogotá), José Fernando Escobar, known for his work on the Yamabe problem, and many more. Proof that a country of 50 million people can punch way above its weight in pure and applied math.