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Kepler's Conjecture: How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World
The fascinating story of a problem that perplexed mathematicians for nearly 400 years
In 1611, Johannes Kepler proposed that the best way to pack spheres as densely as possible was to pile them up in the same way that grocers stack oranges or tomatoes. This proposition, known as Kepler's Conjecture, seemed obvious to everyone except mathematicians, who seldom take anyone's word for anything. In the tradition of Fermat's Enigma, George Szpiro shows how the problem engaged and stymied many men of genius over the centuries--Sir Walter Raleigh, astronomer Tycho Brahe, Sir Isaac Newton, mathematicians C. F. Gauss and David Hilbert, and R. Buckminster Fuller, to name a few--until Thomas Hales of the University of Michigan submitted what seems to be a definitive proof in 1998.
George G. Szpiro (Jerusalem, Israel) is a mathematician turned journalist. He is currently the Israel correspondent for the Swiss daily Neue Zurcher Zeitung.
In 1611, Johannes Kepler proposed that the best way to pack spheres as densely as possible was to pile them up in the same way that grocers stack oranges or tomatoes. This proposition, known as Kepler's Conjecture, seemed obvious to everyone except mathematicians, who seldom take anyone's word for anything. In the tradition of Fermat's Enigma, George Szpiro shows how the problem engaged and stymied many men of genius over the centuries--Sir Walter Raleigh, astronomer Tycho Brahe, Sir Isaac Newton, mathematicians C. F. Gauss and David Hilbert, and R. Buckminster Fuller, to name a few--until Thomas Hales of the University of Michigan submitted what seems to be a definitive proof in 1998.
George G. Szpiro (Jerusalem, Israel) is a mathematician turned journalist. He is currently the Israel correspondent for the Swiss daily Neue Zurcher Zeitung.
304 pages, Hardcover
Published February 14, 2003
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About the author
George G. Szpiro
26 books18 followersGeorge G. Szpiro is an Israeli-Swiss applied mathematician and journalist, who has emerged as a writer of popular science books.
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Displaying 1 - 5 of 5 reviews
September 14, 2015
What's the best way to stack cannonballs on a ship, where space is limited? Sir Walter Raleigh asked a friend of his this question, one Thomas Harriot. He passed the question on to Johannes Kepler, who worked on it for a while, before deciding in exasperation that it was probably the same way melon-sellers stack them in the market. But he couldn't prove it, which got under his, and many other mathematicians', skin. A mere 400 years later, we have the answer.
It turns out Kepler, and the melon-sellers, are correct. But about 300 years after the problem had first been discussed by Kepler, it appeared on David Hilbert's famous lecture in 1900, on the Ten Greatest Problems in mathematics which they should concentrate on solving. In other words, about 300 years after Raleigh asked, mathematicians hadn't really done much except to say, "we think the melon-sellers are right."
In case you're wondering, melon sellers (usually, anyway) stack their melons in a line, then stack the row next to it shifted half a melon, then put the row on top of those two rows in the triangular space between every three melons.
Szpiro's book is a nice mix of history and math, including tidbits like professional rivalries and what wars do to the people who would rather be doing math. There is an extensive diversion through the problem of "kissing spheres", which is how many billiard balls (or other inelastic spheres) can touch a central one simultaneously in two, three, or more dimensions. There are lots of pictures attempting to make the abstruse phraseology of mathematicians more comprehensible. There is a star-studded list of historical characters who touched on this problem (or related ones), including people like Newton and Albrecht Duerer and Lagrange and Hermann Minkowski (now most famous for having called one of his students, Albert Einstein, a "lazy dog", but a distinguished mathematician in his own right).
In the end, the Conjecture was proved by a man named Tom Hales, with help from his graduate students of course, sending out an email announcing this in 1998. In particular, though, this brings up the question of: is this kind of thing really a proof? Part of it involves using a computer program to analyze hundreds or thousands of possible combinations of spheres and the spaces in between them. In the end, did we prove anything?
If the question sounds odd, it is because we have to decide what a proof really is. It's not as if anyone had any serious doubt that this was the most efficient way of packing spheres. Melon makers know their business, after all. The point of a proof, some might say, is to further human understanding. Once we have a proof of the conventional sort, where every detail of it can be written in a paper to be published, which other humans can actually read and understand, we may be prodded to additional questions, or see an extension of this kind of logic to other questions.
Proving a conjecture by using computer power to crank through the possibilities and eliminate them, until only one possibility is left, doesn't really suggest anything of the sort. Also, given that it took many years for Hales and company to do this, will anyone really check if it's true? Similar questions arise with solutions to other long-standing problems, when a computer generates reams of paper that no human can double-check, or even understand.
Which brings us to the deeper question of what it means to prove, or know, anything. Is it enough to have made a machine that, in some fashion, thinks for you, and have that machine prove it? Will you gain anything from that, or is it like using a Cliff's Notes to pass a literature test (the real point of the work eluding you, even if you make the grade you want). But then, how many of us really understand the mathematical proofs behind results we use? I myself waited for the lecturer to get on past the proof part, so they could explain to me how I could use it.
Hard questions, and Szpiro asks them, but does not answer. Which is part of what makes it all a good read, chewy enough to provide grist for your mental mill, but not so much as to choke you. I recommend it.
It turns out Kepler, and the melon-sellers, are correct. But about 300 years after the problem had first been discussed by Kepler, it appeared on David Hilbert's famous lecture in 1900, on the Ten Greatest Problems in mathematics which they should concentrate on solving. In other words, about 300 years after Raleigh asked, mathematicians hadn't really done much except to say, "we think the melon-sellers are right."
In case you're wondering, melon sellers (usually, anyway) stack their melons in a line, then stack the row next to it shifted half a melon, then put the row on top of those two rows in the triangular space between every three melons.
Szpiro's book is a nice mix of history and math, including tidbits like professional rivalries and what wars do to the people who would rather be doing math. There is an extensive diversion through the problem of "kissing spheres", which is how many billiard balls (or other inelastic spheres) can touch a central one simultaneously in two, three, or more dimensions. There are lots of pictures attempting to make the abstruse phraseology of mathematicians more comprehensible. There is a star-studded list of historical characters who touched on this problem (or related ones), including people like Newton and Albrecht Duerer and Lagrange and Hermann Minkowski (now most famous for having called one of his students, Albert Einstein, a "lazy dog", but a distinguished mathematician in his own right).
In the end, the Conjecture was proved by a man named Tom Hales, with help from his graduate students of course, sending out an email announcing this in 1998. In particular, though, this brings up the question of: is this kind of thing really a proof? Part of it involves using a computer program to analyze hundreds or thousands of possible combinations of spheres and the spaces in between them. In the end, did we prove anything?
If the question sounds odd, it is because we have to decide what a proof really is. It's not as if anyone had any serious doubt that this was the most efficient way of packing spheres. Melon makers know their business, after all. The point of a proof, some might say, is to further human understanding. Once we have a proof of the conventional sort, where every detail of it can be written in a paper to be published, which other humans can actually read and understand, we may be prodded to additional questions, or see an extension of this kind of logic to other questions.
Proving a conjecture by using computer power to crank through the possibilities and eliminate them, until only one possibility is left, doesn't really suggest anything of the sort. Also, given that it took many years for Hales and company to do this, will anyone really check if it's true? Similar questions arise with solutions to other long-standing problems, when a computer generates reams of paper that no human can double-check, or even understand.
Which brings us to the deeper question of what it means to prove, or know, anything. Is it enough to have made a machine that, in some fashion, thinks for you, and have that machine prove it? Will you gain anything from that, or is it like using a Cliff's Notes to pass a literature test (the real point of the work eluding you, even if you make the grade you want). But then, how many of us really understand the mathematical proofs behind results we use? I myself waited for the lecturer to get on past the proof part, so they could explain to me how I could use it.
Hard questions, and Szpiro asks them, but does not answer. Which is part of what makes it all a good read, chewy enough to provide grist for your mental mill, but not so much as to choke you. I recommend it.
December 3, 2025
Die Keplersche Vermutung – Mathematik im Computerzeitalter
Das Buch „Die Keplersche Vermutung“ von George G. Szpiro ist eine faszinierende Darstellung eines 400 Jahre alten Problems: der Keplerschen Vermutung, die besagt, dass die dichteste Packung gleich großer Kugeln – wie Orangen in einer Pyramide – durch die kubisch-flächenzentrierte Packung erreicht wird. Der wissenschaftstheoretische Blick richtet sich hierbei kompromisslos auf das „WIE“ der Lösung: eine bahnbrechende methodologische Wende, die die Mathematik ins Computerzeitalter katapultierte.
Die Herausforderung der Unendlichkeit
Jahrhundertelang widerstand die Keplersche Vermutung allen klassischen geometrischen Beweisversuchen, da sie eine unendliche Anzahl möglicher lokaler Kugelarrangements in drei Dimensionen umfasste. Obwohl Carl Friedrich Gauß bereits bewiesen hatte, dass die Vermutung für regelmäßige Gitteranordnungen gilt, blieb der allgemeine Fall – für beliebige, nicht-reguläre Packungen – ungelöst. Die Schwierigkeit bestand darin zu zeigen, dass keine noch so unregelmäßige Anordnung eine höhere Dichte erzielen kann.
Das „WIE“: Die Computer-gestützte Revolution
Die definitive Lösung gelang 1998 dem Mathematiker Thomas Hales, der Mathematik und Informatik auf radikale Weise verband. Sein Ansatz beruhte auf einer massiven Diskretaisierung des kontinuierlichen Problems:
Zerlegung in Fälle: Hales teilte den unendlichen Problemraum in eine endliche, aber immens große Anzahl von 5.000 diskreten Konfigurationen („Standardkonfigurationen“) von Kugeln. Damit reduzierte er das Problem auf die Analyse der Dichte jeder einzelnen Anordnung.
Nicht-lineare Optimierung: Für jede dieser Konfigurationen formulierte Hales das Dichteproblem als Problem nicht-linearer Optimierung und ermittelte rechnerisch die schlechtestmögliche lokale Packungsdichte.
Der Beweis durch Berechnung: Die Überprüfung der unzähligen Ungleichungen und Optimierungsprobleme erforderte Tausende Zeilen komplexer Computersoftware. Der Beweis war damit zu umfangreich, um manuell verifiziert zu werden.
Formale Verifikation
Die Kontroverse um einen Beweis, der vollständig von Maschinen überprüft wurde, führte zum Flyspeck-Projekt. Ziel war die formale Verifikation des gesamten Beweises und des Codes mittels Proof Assistants. 2014 gelang die erfolgreiche formale Überprüfung – ein triumphaler Abschluss der methodologischen Reise. Damit wurde nicht nur die Keplersche Vermutung rational bestätigt, sondern auch die computergestützte Mathematik als legitimes Werkzeug der reinen Mathematik endgültig etabliert – nach dem Vier-Farben-Satz, der als erstes großes mathematisches Problem mithilfe von Computern gelöst wurde, zeigt Hales’ Beweis, dass komplexe, jahrhundertealte Probleme heute auf diese Weise überprüfbar sind.
Das Buch „Die Keplersche Vermutung“ von George G. Szpiro ist eine faszinierende Darstellung eines 400 Jahre alten Problems: der Keplerschen Vermutung, die besagt, dass die dichteste Packung gleich großer Kugeln – wie Orangen in einer Pyramide – durch die kubisch-flächenzentrierte Packung erreicht wird. Der wissenschaftstheoretische Blick richtet sich hierbei kompromisslos auf das „WIE“ der Lösung: eine bahnbrechende methodologische Wende, die die Mathematik ins Computerzeitalter katapultierte.
Die Herausforderung der Unendlichkeit
Jahrhundertelang widerstand die Keplersche Vermutung allen klassischen geometrischen Beweisversuchen, da sie eine unendliche Anzahl möglicher lokaler Kugelarrangements in drei Dimensionen umfasste. Obwohl Carl Friedrich Gauß bereits bewiesen hatte, dass die Vermutung für regelmäßige Gitteranordnungen gilt, blieb der allgemeine Fall – für beliebige, nicht-reguläre Packungen – ungelöst. Die Schwierigkeit bestand darin zu zeigen, dass keine noch so unregelmäßige Anordnung eine höhere Dichte erzielen kann.
Das „WIE“: Die Computer-gestützte Revolution
Die definitive Lösung gelang 1998 dem Mathematiker Thomas Hales, der Mathematik und Informatik auf radikale Weise verband. Sein Ansatz beruhte auf einer massiven Diskretaisierung des kontinuierlichen Problems:
Zerlegung in Fälle: Hales teilte den unendlichen Problemraum in eine endliche, aber immens große Anzahl von 5.000 diskreten Konfigurationen („Standardkonfigurationen“) von Kugeln. Damit reduzierte er das Problem auf die Analyse der Dichte jeder einzelnen Anordnung.
Nicht-lineare Optimierung: Für jede dieser Konfigurationen formulierte Hales das Dichteproblem als Problem nicht-linearer Optimierung und ermittelte rechnerisch die schlechtestmögliche lokale Packungsdichte.
Der Beweis durch Berechnung: Die Überprüfung der unzähligen Ungleichungen und Optimierungsprobleme erforderte Tausende Zeilen komplexer Computersoftware. Der Beweis war damit zu umfangreich, um manuell verifiziert zu werden.
Formale Verifikation
Die Kontroverse um einen Beweis, der vollständig von Maschinen überprüft wurde, führte zum Flyspeck-Projekt. Ziel war die formale Verifikation des gesamten Beweises und des Codes mittels Proof Assistants. 2014 gelang die erfolgreiche formale Überprüfung – ein triumphaler Abschluss der methodologischen Reise. Damit wurde nicht nur die Keplersche Vermutung rational bestätigt, sondern auch die computergestützte Mathematik als legitimes Werkzeug der reinen Mathematik endgültig etabliert – nach dem Vier-Farben-Satz, der als erstes großes mathematisches Problem mithilfe von Computern gelöst wurde, zeigt Hales’ Beweis, dass komplexe, jahrhundertealte Probleme heute auf diese Weise überprüfbar sind.
May 26, 2018
It’s a surprisingly interesting, dramatic story about formal mathematics. It’s mostly accessible to non mathematicians, but not quite which is why I gave it four stars.
Full disclosure: I LOVE math.
Full disclosure: I LOVE math.
October 5, 2023
Extraordinarily math (Heavy)
April 2, 2010
I generally like popular math and physics books. They're some of my favorite non-fiction. Not so Kepler's Conjecture. There's too much academic history here to be of interest to real mathematicians, and too much high-level math, presented too inaccessibly, to be of interest to laymen. To add insult to injury, Szpiro proves in the very early and very late chapters that he's perfectly capable of presenting mathematical material accessibly and charmingly. He just doesn't. The only people I can imagine who might be interested in the book as a whole are prospective graduate students in the field who might be as interested in the stories of scholarly in-fighting as they are in the obscure(d) mathematical proofs.
Displaying 1 - 5 of 5 reviews