Plaster– and string models Jaap Top

Plaster– and string models Jaap Top

IWI-RuG & DIAMANT

use of such models:

‘two’ series from the catalog:

Series VII (Carl Rodenberg)

Series XVII 2a and 2b (Alexander von Brill) and Series XXV (Hermann Wiener)

(note: the 1890 advertisement in American Journal of Math mentions only 16 series)

designer Series XXV:

designer Series XVII 2a and 2b:

designer Series VII:

theme in Series XVII 2a, 2b and Series XXV: plane cubic curves

even today, this subject is widely studied. modern theoretical and practical applications: cryptographic protocols, error cor- recting codes, primality tests and integer factorization methods, diophantine problems, number theory, particles in physics, . . . .

classical question: classify the cubic curves over the real numbers

equivalent problem:

real constants a, b, c, function x 7→ y =

q

x³ + ax² + bx + c

which kind of graphs?

together with reflected graph: all points (x, y) satisfying y² = x³ + ax² + bx + c

such a point (x, y) corresponds to a line ` in R³ containing (0, 0, 0) and (x, y, 1)

union of all these lines is a cone with equation y²z = x³ + ax²z + bxz² + cz³.

the shape of such cones was presented in a string model (Series XXV), or the intersection of the cone with a ball having the origin as center was drawn on that (plaster) ball (Series XVII 2ab)

theory started with Sir Isaac Newton (1643–1727)

appendix Enumeratio Linearum Tertii Ordinis of his book Opticks (1704)

Newton: 5 types depending on the real zeros of p(x) := x³ + ax² + bx + c :

• triple zero. Graph of ±^qp(x):

• double zero and a larger simple one. Graph:

• double zero and a smaller simple one. Graph:

parabola nodata

• only one real zero, and it is simple. Graph:

• three real zeros. Graph:

parabola campaniformis cum ovali

M¨obius: Ueber die Grundformen der Linien der dritten Ordnung

(82 pages, 1852).

Theorem

q

x³ + ax² + bx + c contains:

• no flex points, for the parabola cuspidata and the nodata;

• exactly one flex point in all other cases.

Proof: d²y

dx² = 2p⁰⁰(x)p(x) − (p⁰(x))²

4yp(x) = F (x)

4yp(x) and F⁰(x) = 12p(x).

Now F (x) = 3x⁴ + lower order, so lim

x→±∞F (x) = +∞. Local extrema of F (x) are at point(s) α with p(α) = 0, so F (α) =

−(p⁰(α))². Suppose p(x) has only simple zeros. Then p(x) and p⁰(x) have no zero in common, so F (α) < 0. Hence F (x) has exactly two zeros, and p(x) < 0 at the smaller, p(x) > 0 at the larger one.

Theorem [M¨obius, 1852]

• for parabola nodata and cuspidata, the graph of ^qp(x) con- tains no flex point

• for parabola punctata and campaniformis cum ovali, the tan- gent line at the flex point has positive slope

• there exist three types of parabola pura, namely those with positive/zero/negative slope of the tangent line at the flex point

M¨obius in fact uses a different, equivalent definition:

intersect the cone with a ball as before. The tangent lines at flex points and the vertical line containing the flex points and the plane z = 0 yield large circles on the ball, hence a partition of the ball into triangles and squares.

depending on this partition and whether the curve passes through squares or triangles, one obtains 7 possibilities.

now Series VII, the Rodenberg series: cubic surfaces

theory of cubic surface is considered the starting point of alge- braic geometry

1849, Arthur Cayley & George Salmon

famous example: the ‘diagonal surface’

introduced by Alfred Clebsch (1872)

( v³ + w³ + x³ + y³ + z³ = 0 v + w + x + y + z = 0

contains 27 real lines, 10 Eckardt points, symmetry group S₅, . . .

1866, Alfred Clebsch

Theorem. any smooth cubic surface over C is the closure of ϕ(P²⁽C^{) − {p} , . . . , p }) for certain p , . . . , p ∈ P^2,

example: Clebsch diagonal surface

v = (b − c)(ab + ac − c²) w = ac² + bc² − a³ − c³ x = a(c² − ac − b²)

y = c(a² − ac + bc − b²) z = −v − w − x − y.

modern geometry proof: choose skew lines `, m ⊂ S

consider S → ` × m: P 7→ (Q, R) where P, Q, R are collinear

there are five lines in S meeting ` as well as m, they have a point as image

select one of these five image points (Q, R), ‘blow up’ ` × m in (Q, R) and ‘blow down’ the lines Q × m and ` × R. result: P^2, and inverse is P² → S given by cubic polynomials.

Theorem: this is possible over R if and only if the real surface given by the cubic equation is connected.

classical example: x³ + y³ + z³ + w³ = 0 defines smooth cubic, connected over R^.

Leonhard Euler (1707-1783) parametrized this, but using quartic rather than cubic polynomials:

x = c⁴ − c(a − 3b)(a² + 3b²) y = c(a + 3b)(a² + 3b²) − c⁴ z = (a² + 3b²)² − c³(a + 3b) w = (a − 3b)c³ − (a² + 3b²)²

parametrizations using cubic polynomials were recently found by Noam Elkies and (using the modern proof of Clebsch’s theorem above) by me:

x = −a³ − 2a²c + 3a²b + 12abc − 3ab² − 4ac² + 6b²c + 12bc² + 9b³ y = a³ + 2a²c + 3a²b + 12abc + 3ab² + 4ac² − 6b²c + 12bc² + 9b³ z = −8c³ − 8ac² − 9b³ − a³ − 3a²b − 3ab² − 4a²c − 12b²c

w = 8c³ + 8ac² − 9b³ + a³ − 3a²b + 3ab² + 4a²c + 12b²c.

even in the 21st century, applications of (parametrizing) cubic surfaces were discovered and are studied:

particularly in computer science, computer aided geometric de- sign (CAGD)

recent work of C.L. Bajaj, T.G. Berry, R.L. Holt, S. Lodha, A.N. Netravali, M. Paluszny, R.R. Patterson, J. Warren and oth- ers.

some references

www.math.leidenuniv.nl/~naw/serie5/deel07/jun2006/maanen.pdf (paper on the Mobius classification, in Dutch)

journals.cms.math.ca/cgi-bin/vault/viewprepub/polo8721.prepub (paper on the classification of real cubic surfaces)

dissertations.ub.rug.nl/faculties/science/2007/i.polo.blanco/

(thesis of Irene Polo-Blanco)