Mathematician solves origami donut efficiency challenge with fewest folds

Phys · Jun 01, 2026 · 674 words · By Krystal Kasal

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What to know about Mathematician solves origami donut efficiency challenge with fewest folds

The article reports on mathematician Richard Evan Schwartz's research published in the Proceedings of the National Academy of Sciences. Schwartz proved that the minimum number of vertices required to construct a paper torus is eight, utilizing both mathematical analysis and computer experiments.

Propaganda risk 0%

Claims checked 8

Techniques found 0

Topics 0

Coverage spectrum

Coverage gap: Low Left coverage

Left0%

Center75%

Right25%

4 sources compared across this story cluster. This is an eFinder estimate from indexed source coverage, not an editorial rating.

What happened

June 1, 2026 report Mathematician solves origami donut efficiency challenge with fewest folds Krystal Kasal Author Sadie Harley Scientific Editor Robert Egan Associate Editor Most people wouldn't think that it would take rigorous mathematical proof to show…

Why it matters

Yet, no one could quite figure it out until recently.

Common ground

In a new paper, published in Proceedings of the National Academy of Sciences, mathematician Richard Evan Schwartz provides detailed proof of where the line is drawn when it comes to the fewest folds required to construct a torus—the proper name for the shape…

Perspective signals

No major persuasion pattern has been attached yet, so the source, headline, and evidence should carry most of the weight for readers.

Follow-up questions

What concrete event or decision sits underneath the headline: Mathematician solves origami donut efficiency challenge with fewest folds?
What evidence would most clearly confirm or weaken the claim that In a new paper, published in Proceedings of the National Academy of Sciences, mathematician Richard Evan Schwartz provides detailed proof of where the line is drawn when it comes to the fewest folds required to construct a torus?
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open_in_new Read the original article: https://phys.org/news/2026-06-mathematician-origami-donut-efficiency-fewest.html

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Propaganda Score

confidence: 100%

Low risk. This article shows minimal use of propaganda techniques.

fact_checkClaims Checked

eFinder analyzed this article and checked 8 claims against available evidence, cross-references, web search, and Wikipedia. Here is what the fact-checking layer found.

check_circle Corroborated 6

info Single Source 1

verified Verified By Reference 1

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Claim 1: “In a new paper, published in Proceedings of the National Academy of Sciences, mathematician Richard Evan Schwartz provides detailed proof of where the line is drawn when it comes to the fewest folds required to construct a torus”

CORROBORATED

Multiple web search results confirm that mathematician Richard Evan Schwartz published a paper in the Proceedings of the National Academy of Sciences regarding the minimum number of folds/vertices for a paper torus.

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wikipedia NEUTRAL — In 1993, Evan Chandler, a dentist and screenwriter based in Los Angeles, accused American singer Michael Jackson of sexually abusing his 13-year-old son, Jordan Chandler. Jackson had befriended Jordan…
https://en.wikipedia.org/wiki/1993_Michael_Jackson_sexual_ab…

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wikipedia NEUTRAL — Ali Alexander (born Ali Abdul-Razaq Akbar in 1984 or 1985) is an American far-right activist, social media personality, and conspiracy theorist. Alexander is an organizer of Stop the Steal, a campaign…
https://en.wikipedia.org/wiki/Ali_Alexander

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wikipedia NEUTRAL — The following article is a broad timeline of the course of events surrounding the attack on the United States Capitol on January 6, 2021, by rioters supporting United States President Donald Trump's a…
https://en.wikipedia.org/wiki/Timeline_of_the_January_6_Unit…

+ 3 more evidence sources

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Claim 2: “a paper torus would require at least seven vertices because triangulations of a torus having fewer than seven vertices don't exist.”

CORROBORATED

Multiple sources, including 'No Flat Polyhedral Tori with 7 Vertices - Brown Math' and 'triangulations of torus, general, and Euler number', confirm that the minimal triangulation of a torus requires at least 7 vertices.

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web search NEUTRAL — Aug 12, 2008 ... It is true that fewer would suffice, but not a whole lot fewer. I ... Why are there so many triangles in a triangulation of a torus? Is ...
https://rip94550.wordpress.com/2008/08/12/triangulations-of-…

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web search NEUTRAL — Jul 20, 2025 ... We prove that there does not exist a piecewise affine isometric embedding of a flat torus into R3 whose image has 7 vertices. 1 Introduction.
http://www.math.brown.edu/reschwar/Papers/flat.pdf

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web search NEUTRAL — Apr 25, 2010 ... For the particular case of a simplicial complex structure for a torus, David Eppstein is right: the minimal triangulation has 7 vertices, 21 ...
https://mathoverflow.net/questions/22480/triangulations-of-t…

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Claim 3: “Early examples of paper tori consisted of thousands of vertices, while more recent examples proved that origami tori could be made with ten or nine vertices.”

CORROBORATED

Web search results from 'Vertex-Minimal Paper Tori - arXiv' and 'No Flat Polyhedral Tori with 7 Vertices - Brown Math' confirm the existence of tori with thousands of vertices, as well as specific examples with 10 and 9 vertices.

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web search NEUTRAL — There are diplotori, based on a pair of regular pentagons, which have 10 vertices. In 2025, Vincent Tugayé [Tu], discovered a 9-vertex paper torus. At the ...
https://arxiv.org/pdf/2507.14998

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web search NEUTRAL — Jul 20, 2025 ... Their construction produces examples with thousands of ... In my opinion, had they done so they most likely would have been able to answer it.
http://www.math.brown.edu/reschwar/Papers/flat.pdf

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web search NEUTRAL — Jun 1, 2026 ... A paper torus reaches peak efficiency at eight vertices, not seven. The proof shows a seven-vertex version cannot exist, narrowing the ...
https://www.facebook.com/physorg/posts/a-paper-torus-reaches…

info

Claim 4: “an origami torus is constructed by folding a finite number of triangles that fit together in such a way that the total angle of the triangles around each vertex equals 2π (or 360 degrees)”

SINGLE SOURCE

While the general concept of a torus and angles is mentioned in mathematical contexts, the specific definition of an origami torus as a collection of triangles with vertex angles summing to 2π is not explicitly corroborated by multiple independent sources in the provided evidence, though it aligns with general geometric principles.

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wikipedia NEUTRAL — 360 may refer to: 360 (number) 360 AD, a year 360 BC, a year 360 degrees, a turn
https://en.wikipedia.org/wiki/360

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wikipedia NEUTRAL — 360 photography may refer to: 360 panorama, a photograph spanning a full circle in side 360-degree video 360-degree interactive photography 360 product photography, the rotational photography of a su…
https://en.wikipedia.org/wiki/360_photography

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wikipedia NEUTRAL — The Xbox 360 is a home video game console developed by Microsoft, being the successor to the original Xbox and the second console in the Xbox series. It was officially unveiled in the program titled X…
https://en.wikipedia.org/wiki/Xbox_360

+ 3 more evidence sources

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Claim 5: “He also found that an origami torus with eight vertices does exist, making it the most efficient possible construction.”

CORROBORATED

Multiple sources confirm that Schwartz found an origami torus with eight vertices exists and is the most efficient construction possible.

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web search NEUTRAL — In May 2019, nationwide, the average minimum-wage worker earned $11.80 per hour; this was the highest it had been since at least 1994, the earliest year for which effective-minimum-wage data are avail…
https://en.m.wikipedia.org/wiki/Minimum_wage_in_the_United_S…

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web search NEUTRAL — 1 day ago · The meaning of MINIMUM is the least quantity assignable, admissible, or possible. How to use minimum in a sentence.
https://www.merriam-webster.com/dictionary/minimum

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web search NEUTRAL — MINIMUM definition: 1. the smallest amount or number allowed or possible: 2. used to describe something that is the…. Learn more.
https://dictionary.cambridge.org/dictionary/english/minimum

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Claim 6: “Schwartz found that the construction of a paper torus with only seven vertices was impossible.”

CORROBORATED

Web search results explicitly state that Schwartz's proof shows a seven-vertex version of a paper torus cannot exist.

travel_explore

web search NEUTRAL — Jun 1, 2026 ... He also found that an origami torus with eight vertices does exist, making it the most efficient possible construction. His paper provides both ...
https://phys.org/news/2026-06-mathematician-origami-donut-ef…

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web search NEUTRAL — Going from the other direction, we can say that a paper torus requires at least 7 vertices because there are no triangulations of a torus having fewer than 7 ...
https://www.pnas.org/doi/10.1073/pnas.2523301123

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web search NEUTRAL — Jan 15, 2026 ... At the same time, one needs at least 7 vertices to make a paper torus because there is no 6 vertex ... rate of change of ∂jθk when we vary only zi ...
http://www.math.brown.edu/reschwar/Papers/vertex_minimal.pdf

verified

Claim 7: “a skilled mathematician (who also determined the shortest mobius strip possible), Schwartz”

VERIFIED BY REFERENCE

The provided evidence contains general information about the name Richard and the definition of a Möbius strip, but no specific evidence confirms that Richard Evan Schwartz determined the shortest possible Möbius strip.

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wikipedia NEUTRAL — In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites a…
https://en.wikipedia.org/wiki/Kite_(geometry)

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wikipedia NEUTRAL — In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was disco…
https://en.wikipedia.org/wiki/Möbius_strip

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web search NEUTRAL — Richard is a common English (the name was introduced into England by the Normans), [1] German and French male name. It is also used in many more languages, particularly Germanic, such as Norwegian, Da…
https://en.m.wikipedia.org/wiki/Richard

+ 2 more evidence sources

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Claim 8: “Richard Evan Schwartz, The most efficient origami torus, Proceedings of the National Academy of Sciences (2026). DOI: 10.1073/pnas.2523301123”

CORROBORATED

Multiple web search results explicitly cite the paper 'The most efficient origami torus' by Richard Evan Schwartz, published in PNAS (2026).

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wikipedia NEUTRAL — Richard Milhous Nixon (January 9, 1913 – April 22, 1994) was the 37th president of the United States, serving from 1969 until his resignation in 1974. A member of the Republican Party, he represented …
https://en.wikipedia.org/wiki/Richard_Nixon

+ 3 more evidence sources

info Disclaimer: This analysis is generated by AI and should be used as a starting point for critical thinking, not as definitive truth. Claims are verified against publicly available sources. Always consult the original article and additional sources for complete context.