Book contents
- Frontmatter
- Foreword
- Acknowledgments
- Contents
- The Mystery of the Four-leaf Clovers
- A Fugue
- Tombstone Inscriptions
- The Two Lights
- MMM
- Acquiring Some Personal Items for MMM
- Difficulty in Explaining Relativity Theory in a Few Words
- Difficulty in Obtaining a Cup of Hot Tea
- Hail to Thee, Blithe Spirit
- C. D.
- Cupid's Problem
- The Lighter Life of an Editor
- The Two Kellys
- Some Debts
- Hypnotic Powers
- Founding the Echols Mathematics Club
- Meeting Maurice Fréchet
- Mathematizing the New Mathematics Building
- Finding Some Lost Property Corners
- The Tennessee Valley Authority
- How I First Met Dr. Einstein
- Catching Vibes, and Kindred Matters
- A Pair of Unusual Walking Sticks
- A New Definition
- Dr. Einstein's First Public Address at Princeton
- Parting Advice
- Two Newspaper Items and a Phone Call
- Wherein the Author Is Beasted
- The Scholar's Creed
- The Perfect Game of Solitaire
- The Most Seductive Book Ever Written
- The Master Geometer
- Sandy
- The Perfect Parabola
- Three Coolidge Remarks
- Professor Coolidge during Examinations
- Professor Coolidge's Test
- Borrowing Lecture Techniques from Admired Professors
- My Teaching Assistant Appointment
- A Night in the Widener Memorial Library
- The Slit in the Wall
- Nathan Altshiller Court
- An Editorial Comment
- Intimations of the Future
- A Rival Field
- A Chinese Lesson
- The Bookbag
- Running a Mile in Twenty-one Seconds
- Winning the 1992 Pólya Award
- A Love Story
- Eves' Photo Album
- A Condensed Biography of Howard Eves
- An Abridged Bibliography of Howard Eves' Work
Winning the 1992 Pólya Award
- Frontmatter
- Foreword
- Acknowledgments
- Contents
- The Mystery of the Four-leaf Clovers
- A Fugue
- Tombstone Inscriptions
- The Two Lights
- MMM
- Acquiring Some Personal Items for MMM
- Difficulty in Explaining Relativity Theory in a Few Words
- Difficulty in Obtaining a Cup of Hot Tea
- Hail to Thee, Blithe Spirit
- C. D.
- Cupid's Problem
- The Lighter Life of an Editor
- The Two Kellys
- Some Debts
- Hypnotic Powers
- Founding the Echols Mathematics Club
- Meeting Maurice Fréchet
- Mathematizing the New Mathematics Building
- Finding Some Lost Property Corners
- The Tennessee Valley Authority
- How I First Met Dr. Einstein
- Catching Vibes, and Kindred Matters
- A Pair of Unusual Walking Sticks
- A New Definition
- Dr. Einstein's First Public Address at Princeton
- Parting Advice
- Two Newspaper Items and a Phone Call
- Wherein the Author Is Beasted
- The Scholar's Creed
- The Perfect Game of Solitaire
- The Most Seductive Book Ever Written
- The Master Geometer
- Sandy
- The Perfect Parabola
- Three Coolidge Remarks
- Professor Coolidge during Examinations
- Professor Coolidge's Test
- Borrowing Lecture Techniques from Admired Professors
- My Teaching Assistant Appointment
- A Night in the Widener Memorial Library
- The Slit in the Wall
- Nathan Altshiller Court
- An Editorial Comment
- Intimations of the Future
- A Rival Field
- A Chinese Lesson
- The Bookbag
- Running a Mile in Twenty-one Seconds
- Winning the 1992 Pólya Award
- A Love Story
- Eves' Photo Album
- A Condensed Biography of Howard Eves
- An Abridged Bibliography of Howard Eves' Work
Summary
Two planar pieces that can be placed so that they intercept chords of equal length on each member of some family of parallel lines, or two solid pieces that can be placed so that they intercept sections of equal area on each member of some family of parallel planes, are said to be Cavalieri congruent.
Some years ago I proved the following two theorems which, at first encounter scarcely seem to be true:
Though one can exhibit two tetrahedra of equal volume that are not Cavalieri congruent, any two triangles of equal area are Cavalieri congruent.
Though one can easily show that there exists no polygon Cavalieri congruent to a circle, there exists a tetrahedron Cavalieri congruent to a sphere.
It was my publishing of these two theorems that won me the 1992 Pólya Award. The certificate of the award was accompanied by the following paragraph.
This short article really packs a wallop! The author's two theorems indeed “at first encounter scarcely seem to be true,” and the proofs are excellent illustrations of the power and beauty of geometry. The Cavalieri equivalence of a sphere and a tetrahedron is truly memorable—the sort of result which geometers in ancient times would have inscribed on their tombstones. The article is graced with the author's historical scholarship and lucid prose.
Cavalieri equivalence of a sphere and a tetrahedron
Theorem.There exists a tetrahedron to which a given sphere is Cavalieri congruent.
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- Mathematical Reminiscences , pp. 169 - 170Publisher: Mathematical Association of AmericaPrint publication year: 2001