Author Archives: Michael Weiss

Escher’s Toroidal Print Gallery

If Art+Math brings one person to mind, it’s Escher. His tessellations present the best-known instance, but he did a lot more than that.

In April 2003, the mathematicians Bart de Smit and Hendrik Lenstra wrote a delightful article, Escher and the Droste effect, about Escher’s lithograph Prentententoonstelling. They pointed out that

We shall see that the lithograph can be viewed as drawn on a certain elliptic curve over the field of complex numbers…

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Filed under Analysis, Geometry

Algebraic Geometry Jottings 13

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The Resultant, Episode 3: Inside the Episode

So we have, at long last, several expressions for the resultant:

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Wallpaper Groups

Escher: Alhambra Sketch

I first learned as a kid that “there are only 17 basically different wallpapers” from W.W.Sawyer’s Prelude to Mathematics. (The quote appears on p.102. Aside: this remains an excellent gift for a youngster with a yen for math.) I remember my father pointing out the absurdity of this claim: are all mural wallpapers of van Gogh’s paintings basically the same?

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Filed under Geometry, Physics

Algebraic Geometry Jottings 12

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The Resultant, Episode 3

Last time the linear operator

Φ: Kn[x]⊕Km[x] → Km+n[x]
Φ(p,q)=pE+qF

made its grand entrance, clothed in the Sylvester matrix. (Recall that Kn[x] is the vector space of all polynomials of degree <n with coefficients in K, likewise for Km[x] and Km+n[x].)

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Filed under Algebraic Geometry

Algebraic Geometry Jottings 11

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The Resultant, Episode 2

By now you know the characters: the polynomials E(x) (degree m) and F(x) (degree n) with coefficients in an integral domain R, its fraction field K, and the extension field L of K in which E and F split completely:

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Algebraic Geometry Jottings 10

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The Resultant, Episode 1: Inside the Episode

In Episode 1 of our miniseries, “The Resultant”, the characters were introduced: integral domain R with fraction field K and extension field L, and polynomials E(x) and F(x) in R[x], factoring completely in L as a(x–u1)···(x–um) and b(x–v1)···(x–vn). (Repeated roots allowed.) We had our first formulas for the resultant:

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Filed under Algebraic Geometry

Algebraic Geometry Jottings 9

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The Resultant, Episode 1

Time to discuss the resultant; we’ll need it for Kendig’s proof of Bézout’s theorem, but it has other uses too. The story will take several episodes, plus extras. Like a miniseries!

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Filed under Algebraic Geometry

Algebraic Geometry Jottings 8

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This will be a “chewing” post.

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Algebraic Geometry Jottings 7

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Fulton characterizes the intersection number, I(P, EF), with seven properties. Let me just repeat the last one:

  1.  I(E∩F) = I(E∩(F+AE)) for any A.

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Filed under Algebraic Geometry

Algebraic Geometry Jottings 6

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The tome Commutative Algebra by Zariski and Samuel opens with the memorable sentence, “This book is the child of an unborn parent.” As Zariski explains,

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Filed under Algebraic Geometry