John H. Conway was the John von Neumann Chair of Mathematics at Princeton University. He obtained his BA and his PhD from the University of Cambridge (England). He was a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory, and coding theory. He also contributed to many branches of recreational mathematics, notably the invention of the Game of Life.Heidi Burgiel holds a Ph.D. in Geometry from the University of Washington and a Master’s degree from the Harvard Graduate School of Education. Her professional activities range from mathematical fiber arts through computer systems administration. Over the past 30 years she has worked at the University of Washington, the University of Minnesota, the University of Illinois at Chicago, the Boston Museum of Science, Boston University, Bridgewater State University, the Massachusetts Institute of Technology, Harvard University, the University of Massachusetts at Lowell, and Lasell University.Chaim Goodman-Strauss is Outreach Mathematician at the National Museum of Mathematics (MoMath). Born and raised in Austin, Tex., he earned his Ph.D. in Knot Theory at UT Austin in 1994 and served through 2022 on the mathematics faculty at the University of Arkansas. He has held visiting positions at the Geometry Center at the University of Minnesota, Princeton University, and the Universidad Nacional Autónoma de México.

“The Magic Theorem is a joyful exploration of symmetry and the elegant geometry of orbifolds. Conway, Burgiel, and Goodman-Strauss have created something rare: a book that distills deep mathematics into a playful, visually stunning experience. Whether you're encountering these ideas for the first time or rediscovering them with fresh eyes, this is a guided tour filled with clarity, wonder, and charm.” —Steven Strogatz, Professor of mathematics at Cornell University and bestselling author of Infinite Powers"The present book has a predecessor: The Symmetries of Things, by the same authors, a hefty 400 pages, published in 2008. Conway still worked significantly on this new work, therefore, while his co-authors Heidi Burgiel and Chaim Goodman-Strauss have "much expanded and much abridged" it for this new version, meaning they omitted numerous consequences of the orbifold concept and instead expanded the introduction up to the magic theorem through a wealth of examples. The older book wanted to appeal to "laypeople, artists, active mathematicians, and researchers in general." This new work undoubtedly fulfils this claim as well.As for me, the restriction of the new version to the first part of The Symmetries of Things was actually successful. While I never dared approach the old, much longer book, I was able to consume the new one with profit. And this despite the fact that I haven't been so keen on practice problems since the end of my studies and have skipped the abundantly scattered exercises ("Here is a pattern, find its symmetries")."—Christoph Pöppe, Spektrum der Wissenschaft (translated from the original German article)

• 1 Symmetries. 2 Planar Patterns. 3 The Magic Theorem. 4 Symmetries of Spherical Patterns. 5 The Seven Types of Frieze Patterns. 6 Why the Magic Theorems Work. 7 Euler’s Map Theorem. 8 The Classification of Surfaces. 9 Orbifolds. 10 A Bigger Picture. A Other Notations for the Plane and Spherical Groups.

John H. Conway was the John von Neumann Chair of Mathematics at Princeton University. He obtained his BA and his PhD from the University of Cambridge (England). He was a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory, and coding theory. He also contributed to many branches of recreational mathematics, notably the invention of the Game of Life.Heidi Burgiel holds a Ph.D. in Geometry from the University of Washington and a Master’s degree from the Harvard Graduate School of Education. Her professional activities range from mathematical fiber arts through computer systems administration. Over the past 30 years she has worked at the University of Washington, the University of Minnesota, the University of Illinois at Chicago, the Boston Museum of Science, Boston University, Bridgewater State University, the Massachusetts Institute of Technology, Harvard University, the University of Massachusetts at Lowell, and Lasell University.Chaim Goodman-Strauss is Outreach Mathematician at the National Museum of Mathematics (MoMath). Born and raised in Austin, Tex., he earned his Ph.D. in Knot Theory at UT Austin in 1994 and served through 2022 on the mathematics faculty at the University of Arkansas. He has held visiting positions at the Geometry Center at the University of Minnesota, Princeton University, and the Universidad Nacional Autónoma de México.

“The Magic Theorem is a joyful exploration of symmetry and the elegant geometry of orbifolds. Conway, Burgiel, and Goodman-Strauss have created something rare: a book that distills deep mathematics into a playful, visually stunning experience. Whether you're encountering these ideas for the first time or rediscovering them with fresh eyes, this is a guided tour filled with clarity, wonder, and charm.” —Steven Strogatz, Professor of mathematics at Cornell University and bestselling author of Infinite Powers"The present book has a predecessor: The Symmetries of Things, by the same authors, a hefty 400 pages, published in 2008. Conway still worked significantly on this new work, therefore, while his co-authors Heidi Burgiel and Chaim Goodman-Strauss have "much expanded and much abridged" it for this new version, meaning they omitted numerous consequences of the orbifold concept and instead expanded the introduction up to the magic theorem through a wealth of examples. The older book wanted to appeal to "laypeople, artists, active mathematicians, and researchers in general." This new work undoubtedly fulfils this claim as well.As for me, the restriction of the new version to the first part of The Symmetries of Things was actually successful. While I never dared approach the old, much longer book, I was able to consume the new one with profit. And this despite the fact that I haven't been so keen on practice problems since the end of my studies and have skipped the abundantly scattered exercises ("Here is a pattern, find its symmetries")."—Christoph Pöppe, Spektrum der Wissenschaft (translated from the original German article)

• 1 Symmetries. 2 Planar Patterns. 3 The Magic Theorem. 4 Symmetries of Spherical Patterns. 5 The Seven Types of Frieze Patterns. 6 Why the Magic Theorems Work. 7 Euler’s Map Theorem. 8 The Classification of Surfaces. 9 Orbifolds. 10 A Bigger Picture. A Other Notations for the Plane and Spherical Groups.

John H. Conway was the John von Neumann Chair of Mathematics at Princeton University. He obtained his BA and his PhD from the University of Cambridge (England). He was a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory, and coding theory. He also contributed to many branches of recreational mathematics, notably the invention of the Game of Life.Heidi Burgiel holds a Ph.D. in Geometry from the University of Washington and a Master’s degree from the Harvard Graduate School of Education. Her professional activities range from mathematical fiber arts through computer systems administration. Over the past 30 years she has worked at the University of Washington, the University of Minnesota, the University of Illinois at Chicago, the Boston Museum of Science, Boston University, Bridgewater State University, the Massachusetts Institute of Technology, Harvard University, the University of Massachusetts at Lowell, and Lasell University.Chaim Goodman-Strauss is Outreach Mathematician at the National Museum of Mathematics (MoMath). Born and raised in Austin, Tex., he earned his Ph.D. in Knot Theory at UT Austin in 1994 and served through 2022 on the mathematics faculty at the University of Arkansas. He has held visiting positions at the Geometry Center at the University of Minnesota, Princeton University, and the Universidad Nacional Autónoma de México.

“The Magic Theorem is a joyful exploration of symmetry and the elegant geometry of orbifolds. Conway, Burgiel, and Goodman-Strauss have created something rare: a book that distills deep mathematics into a playful, visually stunning experience. Whether you're encountering these ideas for the first time or rediscovering them with fresh eyes, this is a guided tour filled with clarity, wonder, and charm.” —Steven Strogatz, Professor of mathematics at Cornell University and bestselling author of Infinite Powers"The present book has a predecessor: The Symmetries of Things, by the same authors, a hefty 400 pages, published in 2008. Conway still worked significantly on this new work, therefore, while his co-authors Heidi Burgiel and Chaim Goodman-Strauss have "much expanded and much abridged" it for this new version, meaning they omitted numerous consequences of the orbifold concept and instead expanded the introduction up to the magic theorem through a wealth of examples. The older book wanted to appeal to "laypeople, artists, active mathematicians, and researchers in general." This new work undoubtedly fulfils this claim as well.As for me, the restriction of the new version to the first part of The Symmetries of Things was actually successful. While I never dared approach the old, much longer book, I was able to consume the new one with profit. And this despite the fact that I haven't been so keen on practice problems since the end of my studies and have skipped the abundantly scattered exercises ("Here is a pattern, find its symmetries")."—Christoph Pöppe, Spektrum der Wissenschaft (translated from the original German article)

• 1 Symmetries. 2 Planar Patterns. 3 The Magic Theorem. 4 Symmetries of Spherical Patterns. 5 The Seven Types of Frieze Patterns. 6 Why the Magic Theorems Work. 7 Euler’s Map Theorem. 8 The Classification of Surfaces. 9 Orbifolds. 10 A Bigger Picture. A Other Notations for the Plane and Spherical Groups.