Henry Segerman
Department of Mathematics
Oklahoma State University
Stillwater, OK 74078, USA
Office: 504 Mathematical Sciences
Phone: (405) 744-7746
Email: henry (dot) segerman (at) okstate (dot) edu
Teaching
Office hours for Spring 2024: MTW TBD. Monday and Wednesday will be via Zoom (see the syllabus on Canvas for the link to the Zoom meeting). Tuesday will be in MSCS 421.
Webpages
Book
Visualizing Mathematics with 3D Printing
Published by Johns Hopkins University Press, July 2016, 186 pages, 150 figures. This is a popular mathematics book, with the twist that most of the figures in the book are photographs of 3D printed models. These models are available for readers to download and 3D print themselves, or order online, or explore virtually on the book's website 3dprintmath.com.
Papers and preprints
Geometry and Topology
Connecting essential triangulations I: via 2-3 and 0-2 moves, arXiv: 2405.03539 [math.GT], with Tejas Kalelkar and Saul Schleimer, 57 pages, 74 figures and subfigures.
Non-standard bi-orders on punctured torus bundles, arXiv:2310.13758 [math.GT, math.GR], with Jonathan Johnson, 27 pages, 4 figures.
From veering triangulations to dynamic pairs, arXiv:2305.08799 [math.GT], with Saul Schleimer, 77 pages, 90 figures and subfigures.
From veering triangulations to link spaces and back again, arXiv:1911.00006 [math.GT], with Steven Frankel and Saul Schleimer, 127 pages, 99 figures and subfigures.
From loom spaces to veering triangulations, Groups, Geometry, and Dynamics 18 (2024), no. 2, 419-462, with Saul Schleimer, 44 pages, 31 figures and subfigures.
Ray-marching Thurston geometries, with Rémi Coulon, Elisabetta A. Matsumoto, and Steve J. Trettel, Experimental Mathematics 31 (2022), no. 4, 1197-1277, 81 pages, 198 figures and subfigures.
Cohomology fractals, Cannon-Thurston maps, and the geodesic flow, with David Bachman, Matthias Goerner, and Saul Schleimer, Experimental Mathematics 31 (2022), no. 4, 1047-1085, 39 pages, 80 figures and subfigures.
Essential loops in taut ideal triangulations, with Saul Schleimer, Algebraic and Geometric Topology 20 (2020), no. 1, 487-501, 14 pages, 6 figures.
Traversing three-manifold triangulations and spines, with J. Hyam Rubinstein and Stephan Tillmann, L'Ensignement Mathématique, 65 (2019), no. 2, 155-206, 41 pages, 42 figures.
Connectivity of triangulations without degree one edges under 2-3 and 3-2 moves, Proc. Amer. Math. Soc., 145 (2017), no. 12, pp. 5391-5404, 14 pages, 15 figures.
Non-geometric veering triangulations, with Craig D. Hodgson and Ahmad Issa, Experimental Mathematics 25 (2016), no. 1, pp. 17-45, 29 pages, 24 figures.
Triangulations of 3-manifolds with essential edges, with Craig D. Hodgson, J. Hyam Rubinstein, and Stephan Tillmann, Annales de Mathématiques de Toulouse 24 (2015) no. 5, pp. 1103-1145, 43 pages, 14 figures.
1-efficient triangulations and the index of a cusped hyperbolic 3-manifold, with Stavros Garoufalidis, Craig D. Hodgson and J. Hyam Rubinstein, Geometry & Topology 19 (2015), pp. 2619--2689, 71 pages, 28 figures.
Triangulations of hyperbolic 3-manifolds admitting strict angle structures, with Craig D. Hodgson and J. Hyam Rubinstein, Journal of Topology 5 (2012), no. 5, pp. 887-908, 22 pages, 9 figures.
A generalisation of the deformation variety, Algebraic and Geometric Topology 12 (2012), no. 4, pp. 2179-2244, 66 pages, 26 figures.
Pseudo-developing maps for ideal triangulations I: Essential edges and generalised hyperbolic gluing equations, with Stephan Tillmann, Topology and Geometry in Dimension Three: Triangulations, Invariants, and Geometric Structures (Proceedings of the Jacofest conference), AMS Contemporary Mathematics 560 (2011), pp. 85-102, 18 pages, 8 figures.
Veering triangulations admit strict angle structures, with Craig D. Hodgson, J. Hyam Rubinstein and Stephan Tillmann, Geometry & Topology 15 (2011), pp. 2073-2089, 17 pages, 9 figures.
Incompressible surfaces in handlebodies and boundary compressible 3-manifolds, with J. Nogueira, Topology and its Applications 158 (2011), no. 4, pp. 551-571, 21 pages, 14 figures.
Detection of incompressible surfaces in hyperbolic punctured torus bundles, Geometriae Dedicata 150 (2011), no. 1, pp. 181-232, 52 pages, 25 figures.
On spun-normal and twisted squares surfaces, Proc. Amer. Math. Soc. 137 (2009), pp. 4259-4273, 15 pages, 13 figures.
Mechanisms
A mathematical overview and some applications of gear design, with Elisabetta A. Matsumoto, 3D Printing in Mathematics, Proceedings of Symposia in Applied Mathematics, Volume 79, (2023), pp. 1-17. 18 pages, 48 figures and subfigures.
Self-Similar Quadrilateral Tilings and Deployable Scissor Grids, with Kyle VanDeventer, Proceedings of Bridges 2022: Mathematics, Music, Art, Architecture, Culture, pp. 313-316, 4 pages, 19 figures and subfigures.
Conjugate shape simplification via precise algebraic planar sweeps, with Gershon Elber and Jinesh Machchhar, Computers & Graphics 90 (2020), pp. 1-10, 10 pages, 10 figures.
Geared Jitterbugs, with Elisabetta A. Matsumoto, Proceedings of Bridges 2019: Mathematics, Music, Art, Architecture, Culture, pp. 399-402, 4 pages, 9 figures.
Triple gear, with Saul Schleimer, Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture, pp. 353-360, 8 pages, 19 figures.
Mathematical Exposition, Visualization, and Art
Surfaces in the Tesseract, with Manuel Estévez and Érika Roldán, Proceedings of Bridges 2023: Mathematics, Music, Art, Architecture, Culture, pp. 441-444, 4 pages, 20 figures and subfigures.
Ars Mathemalchemica: From Math to Art and Back Again, with Susan Goldstine and Elizabeth Paley, Notices of the American Mathematical Society 69 (2022), no. 7, pp. 1220-1229, 10 pages, 26 figures and subfigures.
The art of illustrating mathematics, with Edmund Harriss, Journal of Mathematics and the Arts 16 (2022), nos. 1-2, pp. 1-10, 10 pages, 3 figures.
Rolling acrobatic apparatus, Notices of the American Mathematical Society 68 (2021), no. 7, pp. 1106-1118, 13 pages, 49 figures and subfigures.
Cohomology fractals, with David Bachman and Saul Schleimer, Proceedings of Bridges 2020: Mathematics, Music, Art, Architecture, Culture, pp. 175-182, 8 pages, 30 figures and subfigures.
Non-euclidean virtual reality III: Nil, with Rémi Coulon, Elisabetta A. Matsumoto, and Steve Trettel, Proceedings of Bridges 2020: Mathematics, Music, Art, Architecture, Culture, pp. 153-160, 8 pages, 33 figures and subfigures.
Non-euclidean virtual reality IV: Sol, with Rémi Coulon, Elisabetta A. Matsumoto, and Steve Trettel, Proceedings of Bridges 2020: Mathematics, Music, Art, Architecture, Culture, pp. 161-168, 8 pages, 20 figures and subfigures.
Möbius Cellular Automata Scarves, with Elisabetta A. Matsumoto and Fabienne Serriere, Proceedings of Bridges 2018: Mathematics, Music, Art, Architecture, Culture, pp. 523-526, 4 pages, 6 figures.
Conformally correct tilings, published as Pavages effectivement conformes with Saul Schleimer, Objets mathématiques (2017), CNRS Editions, pp. 140-147, 6 pages, 7 compound figures.
Numerically Balanced Dice, with Robert Bosch and Robert Fathauer, The Mathematics of Various Entertaining Subjects: Research in Games, Graphs, Counting, and Complexity, Volume 2 (2017), pp. 253-268, 16 pages, 10 figures.
Non-euclidean virtual reality I: explorations of H^{3}, with Vi Hart, Andrea Hawksley, and Elisabetta A. Matsumoto, Proceedings of Bridges 2017: Mathematics, Music, Art, Architecture, Culture, pp. 33-40, 8 pages, 9 figures.
Non-euclidean virtual reality II: explorations of H^{2}×E, with Vi Hart, Andrea Hawksley, and Elisabetta A. Matsumoto, Proceedings of Bridges 2017: Mathematics, Music, Art, Architecture, Culture, pp. 41-48, 8 pages, 7 figures.
Magnetic sphere constructions, with Rosa Zwier, Proceedings of Bridges 2017: Mathematics, Music, Art, Architecture, Culture, pp. 79-86, 8 pages, 10 figures.
Visualizing Hyperbolic Honeycombs, with Roice Nelson, Journal of Mathematics and the Arts 11 (2017), no. 1, pp. 4-39, 36 pages, many figures.
Squares that Look Round: Transforming Spherical Images, with Saul Schleimer, Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Culture, pp. 15-24, 10 pages, 9 figures with many subfigures.
Puzzling the 120-cell, with Saul Schleimer, Notices of the American Mathematical Society 62 (2015), no. 11, pp. 1309-1316, 8 pages, many figures. A more detailed version is available at arXiv:1310.3549 [math.GT].
Hypernom: Mapping VR Headset Orientation to S^{3}, with Vi Hart, Andrea Hawksley and Marc ten Bosch, Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture, pp. 387-390, 4 pages, 3 figures.
The Quaternion Group as a Symmetry Group, with Vi Hart, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture, pp. 143-150, 8 pages, 8 figures. This paper was republished in The Best Writing on Mathematics 2015, Princeton University Press.
Developing fractal curves, with Geoffrey Irving, Journal of Mathematics and the Arts 7 (2013), no. 3-4, pp. 103-121. 21 pages, 22 figures.
How to print a hypercube, Math Horizons 20 (Feb 2013), issue 3, pp. 5-9, 5 pages, 9 figures.
3D printing for mathematical visualisation, Math. Intell. 34 (2012), no. 4, pp. 56-62, 7 pages, 9 figures. The final publication is available at www.springerlink.com, at this page.
Sculptures in S^{3}, with Saul Schleimer, Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture, pp. 103-110, 8 pages, 9 figures.
Recent 3D printed sculptures, Hyperseeing 2011 Fall/Winter, 10 pages, 11 figures.
Fractal graphs by iterated substitution, Journal of Mathematics and the Arts (© Taylor and Francis, available online at: http://www.tandfonline.com), 5 (2011), no. 2, pp. 51-70, 20 pages, 20 figures.
The Sunflower Spiral and the Fibonacci metric, Proceedings of Bridges 2010: Mathematics, Music, Art, Architecture, Culture (2010), pp. 483-486, 4 pages, 4 figures.
Autologlyphs, with Paul-Olivier Dehaye, 3 pages, many figures. Mathematical Intelligencer 26 (2004), no. 2, pp. 37-39, and cover art.
100 prisoners and a lightbulb, with Paul-Olivier Dehaye and Daniel Ford, Mathematical Intelligencer 25 (2003), no. 4, pp. 53-61, 9 pages, 3 figures.
Reviews
Shaped to roll along a programmed periodic path, with Elisabetta Matsumoto, Nature 620 (2023), pp. 282-283, 2 pages, 1 figure.
Hyperbolic Knot Theory, by Jessica S. Purcell, MAA Reviews 4/4/2021.
My Ph.D. thesis is available here (submitted May 2007). It bears a striking resemblance to the paper "Detection of incompressible surfaces in hyperbolic punctured torus bundles".
I designed the cover art and all illustrations for the book Blast into Math! by my friend Julie Rowlett, published January 2013.
Talks
Slides and notes from some of my talks (see my CV for more):
- From veering triangulations to pseudo-Anosov flows (pdf/keynote), December 2018.
- 3D Shadows: Casting Light on the Fourth Dimension, March 2017.
- Visualizing Mathematics with 3D Printing, March 2017.
- Navigating the three-sphere via quaternions, January 2017.
- Design by transformation, January 2017.
- Using Rhinoceros and Python to produce 3D printed mathematical models - workshop notes (last updated July 2016).
- 3D Printing in Mathematics, October 2015.
- Hypernom: Mapping VR Headset Orientation to S^{3}, July 2015, Link to video, Link to spherical video.
- Veering Dehn surgery, March 2015.
- Rep-tiles, fractal curves, 3D printing and the 4th dimension, October 2014, Link to video.
- The quaternion group as a symmetry group, August 2014, Link to video.
- Developing fractal curves, August 2014, Link to video.
- Sculpture in four-dimensions, June 2014, Link to video.
- Design of 3D printed mathematical art, June 2014, Link to video.
- Structure on the set of triangulations, June 2014.
- How to make sculptures of 4-dimensional things, April 2014, Link to video.
- Regular triangulations and the index of a cusped hyperbolic 3-manifold, October 2013.
- Puzzling the 120-cell, August 2013, Link to video.
- Triple gear, July 2013, Link to video.
- Fractal curves, 4-dimensional puzzles and unlikely gears, April 2013.
- Fractals and how to make a Sierpinski Tetrahedron, November 2012. The activity following this talk is to build a large Sierpinski tetrahedron from 256 small tetrahedra, which can be made from 128 copies of this page.
- Sculptures in S^{3}, July 2012, Link to video.
- Some Mathematical Sculptures, January 2012.
- Triangulations of hyperbolic 3-manifolds admitting strict angle structures, January 2012.
- A generalisation of the deformation variety, Oct 2010.
- The Sunflower Spiral and the Fibonacci Metric, Jul 2010.
- When is a knot not a knot?, Jan 2010.
- The Mathfest 2009 Poster Image, Mathematical Art, Design and Education in Second Life, Aug 2009.
I also showed a video at the start of the talk and another at the end. - Drawing knots using computers, Jul 2009.
- Extending the deformation variety, Nov 2008.
- Ideal triangulations and components of the Character variety, Nov 2007.
On 23rd February 2008 I gave a talk to the University of Texas Mathematics Department's Saturday Morning Math Group (for an audience of around 200 high school students from nearby schools in the area), you can see a video of it here (scroll down to my talk).
Art Exhibitions
- Art by Number, Peninsula School of Art, Door County, WI, October 9 -- November 27, 2021, (2,3,5) triangle tiling (with Saul Schleimer); Stereographic projection (grid).
- For the Love of Math, Suzanne Zahr gallery, Mercer Island, WA, October 1 -- November 24 2021, Cohomology Fractals (with David Bachman and Saul Schleimer), 15+4 Puzzle, and Hyperbolic 29 Puzzle.
- Mathematics: Vintage and Modern, SFO Museum, San Francisco International Airport, San Francisco, CA, July 30, 2021 -- May 01, 2022, Developing Terdragon Curve and Developing Sierpinski Arrowhead Curve.
- Bridges conference 2021, Online, August 2021: Jupiter Gyroid (with Jason Krugman and Sabetta Matsumoto), and 15 + 4 puzzle.
- Joint Mathematics Meetings 2021, Online, January 2021: Closed cohomology fractals (with David Bachman, Matthias Goerner, and Saul Schleimer).
- Bridges Conference 2020, Aalto University, Helsinki, Finland, August 2020 (moved online due to COVID-19): Cohomology fractals (with David Bachman and Saul Schleimer).
- ICERM Illustrating Mathematics semester program, Providence, RI, September 4-December 6, 2019: (3,3) Seifert Surface and (3,3) Seifert Surface with Fibers (with Saul Schleimer), Cannon-Thurston Maps (with David Bachman and Saul Schleimer), Figure-eight knot complement (with François Guéritaud and Saul Schleimer), and Möbius Cellular Automata Scarves} (with Sabetta Matsumoto and Fabienne Serriere).
- Bridges conference 2019, Linz, Austria, July 16-19, 2019: Geared Jitterbugs (with Sabetta Matsumoto), and in the short movie festival: Non-euclidean virtual reality using ray marching.
- Mathapalooza!, Decatur, GA, March 9, 2019: Tetrahedral racks (with Saul Schleimer) and Developing Hilbert Curve.
- Joint Mathematics Meetings 2019, Baltimore, MD, January 16-19, 2019: Cannon-Thurston Ball (with David Bachman), and Five axis racks (with Sabetta Matsumoto).
- Bridges conference 2018 Nominee's Gallery: Möbius Cellular Automata Scarves (with Sabetta Matsumoto and Fabienne Serriere), and in the short movie festival: Peace for Triple Piano and The Making of "Peace for Triple Piano" (both with Vi Hart).
- Joint Mathematics Meetings 2018: Trefoil spine (with Saul Schleimer), and Juggling motion.
- Bridges conference 2017 General Exhibition: Tetrahedral racks and Borromean racks (both with Saul Schleimer).
- Brilliant Geometry: An interactive exhibition, Summerhall, Edinburgh, UK, 13 May - 4 June 2017, solo exhibition with Saul Schleimer, Peter Reid, Mark Reynolds and Sabetta Matsumoto.
- Joint Mathematics Meetings 2017: Tetrahedral racks and Borromean racks (with Saul Schleimer).
- Bridges conference 2016: Klein Quartic, (with Saul Schleimer), and in the short movie festival: Spherical Droste video, and Illuminating hyperbolic geometry (with Saul Schleimer).
- Illustrating Mathematics, Mathematical Sciences Research Institute, 10 pieces (with Saul Schleimer and Will Segerman), February 25 -- March 6, 2016.
- Joint Mathematics Meetings 2016: Klein Quartic and (2,3,5) triangle tiling (with Saul Schleimer), Stereographic projection (grid).
- In the Realm of Forms, Pearl Conard Art Gallery, Ohio State University, Mansfield Campus, OH: Conformal Chmutov, Seifert surface for (2,2) torus link, Seifert surface for (2,2) torus link with fibers, Seifert surface for (3,3) torus link, Seifert surface for (3,3) torus link with fibers (all with Saul Schleimer), November 9 -- December 8, 2015.
- MathThematic: a fine art exhibition, Esther Klein Gallery, Philadelphia, PA: Quaternary Tree Mobile (Level 5), (with Marco Mahler), October 7 -- November 20, 2015.
- Bridges conference 2015: Hypernom (with Vi Hart, Andrea Hawksley and Marc ten Bosch), Monkey See, Monkey Do (with Vi Hart, Andrea Hawksley, Will Segerman and Marc ten Bosch), Hyperbolic Catacombs and {∞,∞,∞} (with Roice Nelson), and in the short movie festival, Torus knots and Seifert surfaces (with Saul Schleimer).
- Joint Mathematics Meetings 2015: Hilbert Sphere (with David Bachman and Robert Fathauer), Hyperbolic Catacombs and Regular {4,6,4} H^{3} Honeycomb (with Roice Nelson), and More fun than a hypercube of monkeys (with Will Segerman).
- Bridges conference 2014: Developing Fractal Curves, More fun than a hypercube of monkeys (with Will Segerman), "Buckyball" Buckyball (with Rosa Zwier).
- Simons Center for Geometry and Physics, Illustrating Geometry Art Exhibition, solo show with Saul Schleimer, 19 June - 1 August 2014. Catalog (26 pieces), Posters, Walkthrough video.
- Joint Mathematics Meetings 2014: Triple gear (with Saul Schleimer).
- First Annual Museum of Mathematics Gala, October 2013, 49 colour 3D prints designed as table centerpieces.
- Bridges conference 2013: Triple gear (with Saul Schleimer; won the "Most Innovative" People's Choice Award, one of four awards given), Seifert surfaces for (3,3) and (4,4) torus links (with Saul Schleimer), "Bunny" Bunny (with Craig S. Kaplan).
- Edinburgh International Science Festival, City Art Centre Exhibition 2013: ten sculptures.
- Joint Mathematics Meetings 2013: Seifert surfaces for torus knots and links (with Saul Schleimer), Developing Fractal Curves.
- Bridges conference 2012: Dual Half 120- and 600-Cells (with Saul Schleimer; won the "Most Effective Use of Mathematics" People's Choice Award, one of four awards given.)
- Joint Mathematics Meetings 2012: Round Möbius strip, Round Klein bottle (with Saul Schleimer).
- Bridges conference 2011: Space filling graph 1, Octahedron fractal graph, Cuboctahedral fractal graph.
- Bridges conference 2010: Sphere autologlyph, Torus autologlyph.
Video
My YouTube channel has over 20 million views.
Guest appearances on the Numberphile, StandUpMaths, and Vi Hart channels on YouTube.
Press
1 January 2023: Siobhan Roberts wrote an article about my puzzles and mechanisms for the New York Times.
23 September 2020: Alison Henrich interviewed me for Mathematics Magazine.
27 August 2017: Samantha Quick produced a spherical video for the New York Times "Daily 360", that I made with M Eifler, Vi Hart, Andrea Hawksley and Elisabetta Matsumoto, based on our work on non-euclidean virtual reality.
20 May 2017: James Glossop wrote on the UK Times website about a photograph he took of me at our Brilliant Geometry exhibition in Edinburgh.
21 March 2017: Davide Castelvecchi wrote an article for Nature News on my work with Vi Hart, Andrea Hawksley and Sabetta Matsumoto on non-euclidean virtual reality. The article appeared in print: Nature 543, p 473 (23 March 2017)
15 November 2016: Luke Whelan wrote an article for Wired on my work in 3D printing and virtual reality, and my book Visualizing Mathematics with 3D Printing.
June 2016: Roman Fishman (Роман Фишман) wrote a feature about my 3D printed artwork that was published in the June 2016 print edition of Popular Mechanics (Russia). The article appeared online on July 18 2016.
26 April 2016: Siobhan Roberts wrote an article for The New Yorker, on my work with Robert Fathauer and Bob Bosch on making the first injection moulded 120-sided die. Other major press included Liz Stinson writing for Wired, Holger Dambeck writing for Speigel Online, and a video by Karin Heineman for Inside Science.
1 February 2016: Evelyn Lamb wrote a blog post for the American Mathematical Society, on my investigation into editing spherical video with Möbius transformations.
14 December 2015: Gary Antonick featured a 3D animation of Saul Schleimer's and my Triple gear at the end of a New York Times Numberplay post.
15 February 2015: I gave a talk on visualising four-dimensional objects at the annual meeting of the American Association for the Advancement of Science. Science, New Scientist and NBC News wrote blog posts about it.
30 October 2014: Alex Bellos wrote a blog post for The Guardian entitled "Pumpkin geometry: stunning shadow sculptures that illuminate an ancient mathematical technique", on stereographic projection sculptures by Saul Schleimer and me.
19 May 2014: Evelyn Lamb wrote a blog post for Scientific American on "More fun than a hypercube of monkeys", a sculpture by my brother Will and me, inspired by a question of Vi Hart.
15 March 2013: Megan Gambino wrote a blog post for Smithsonian magazine on my 3D printed sculpture.
31 October 2012: Evelyn Lamb wrote a blog post for Scientific American on the "30-cell puzzle", a 3D printed puzzle by Saul Schleimer and me based on the 120-cell, one of the regular 4-dimensional polytopes.
23 August 2012: My sculpture "Dual Half 120- and 600-Cells" (joint work with Saul Schleimer) won the "Best Use of Mathematics" prize (one of four prizes awarded) at the Bridges conference 2012. It was featured in a slide show on the Scientific American website.
January 2012: I was interviewed for an article in the "Voice", a publication put out by the University of Melbourne, on "The language and art of maths". Here is the article. The online version of the article is missing the image (shown here) that went with the article. Alternatively, here is a pdf of the whole publication, which includes the article (on page 4) together with the photograph.
6 December 2011: I was interviewed by BBC Radio 5 about mathematical art and 3D printing. Here is the blogpost for the podcast. Here is the direct link to the mp3.
February 2008: I was interviewed on She Blinded Me With Science!, a show on the student radio station KVRX, talking about topology and juggling. You can listen to the interview here.
Links
I design and produce mathematically interesting dice with Robert Fathauer, see The Dice Lab.
Go here for my personal website. Particularly math(s) related things: 3D printed sculpture, Autologlyphs, Escher's Printgallery at Stanford, Book Covers and Posters, T-shirt designs.
For keeping track of your and your friends' (lack of) mathematical progress: Ways to Go Wrong Tally Sheet.
My old University of Melbourne math website is here. My old University of Texas math website is here. My old Stanford math website is here.
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