Stewart Toroid Record: New 291-Genus Holy Monster Explained
There has been significant progress in the search for the Stewart Toroid of the highest genus, also known as the "Holy Monster," a shape with the maximum possible number of holes.
The Original Holy Monster
The concept originates from Bonnie Stewart's book, Adventures Among the Toroids. Stewart's original Holy Monster, a shape with 46 holes (genus 46), was considered the best possible at the time. This shape had 934 faces.
Robert Webb's Contribution
Robert Webb, creator of the software Stella, discovered a new polyhedron. He then ingeniously placed Stewart's original Holy Monster inside his new polyhedron and joined them. This combination resulted in a shape with a genus of 87.
The New Record: Genus 291
Recently, the record for the highest genus has dramatically increased. New toroids have been created with a genus of 291. One such example uses Robert Webb's original toroid as its outer structure, identifiable by its mauve faces. However, the internal structure is far more complex than Stewart's original Holy Monster.
This internal polyhedron, while riddled with holes, would not meet the criteria for a Stewart Toroid on its own because it is not quasi-convex. However, by enclosing it within Webb's original polyhedron, the entire compound object becomes quasi-convex, achieving the record-breaking 291 holes.
The Creator
This latest advancement was made by an individual known as "Squilliams," who shared their work on a polyhedra enthusiast board.
Takeaways
- The original "Holy Monster" described by Bonnie Stewart had 46 holes (genus 46) and 934 faces, representing the highest known genus at its time.
- Robert Webb later combined Stewart's original toroid with his own newly discovered polyhedron, creating a compound shape with a genus of 87.
- A recent breakthrough by a contributor known as Squilliams produced a compound toroid with a record‑breaking genus of 291 holes.
- The new 291‑genus toroid uses Webb’s original toroid as an outer shell while the inner structure, though riddled with holes, is not quasi‑convex on its own.
- By enclosing the non‑quasi‑convex inner polyhedron within Webb’s outer shell, the entire compound becomes quasi‑convex, satisfying the criteria for a Stewart Toroid and achieving the new record.
Frequently Asked Questions
What does 'quasi-convex' mean in the context of Stewart Toroids?
Quasi‑convex means that for any two points on the surface, the line segment connecting them lies entirely inside the polyhedron or on its surface, allowing some indentations but no interior cavities that break convexity. In Stewart Toroids, quasi‑convexity ensures the shape can be considered a single toroidal object despite many holes.
How does enclosing a non‑quasi‑convex polyhedron inside Webb's toroid make the compound quasi‑convex?
Enclosing the non‑quasi‑convex inner polyhedron inside Webb’s outer toroid fills the interior gaps that would otherwise violate quasi‑convexity, because the outer shell provides a continuous surface that contains all interior points. The combined object therefore meets the quasi‑convex definition, allowing it to qualify as a Stewart Toroid.
Who is Numberphile on YouTube?
Numberphile is a YouTube channel that publishes videos on a range of topics. Browse more summaries from this channel below.
Does this page include the full transcript of the video?
Yes, the full transcript for this video is available on this page. Click 'Show transcript' in the sidebar to read it.
Helpful resources related to this video
If you want to practice or explore the concepts discussed in the video, these commonly used tools may help.
Links may be affiliate links. We only include resources that are genuinely relevant to the topic.