Solving Cubics With Creases: The Work of Beloch and Lill

@article{Hull2011SolvingCW,
  title={Solving Cubics With Creases: The Work of Beloch and Lill},
  author={Thomas C. Hull},
  journal={The American Mathematical Monthly},
  year={2011},
  volume={118},
  pages={307 - 315}
}
  • Thomas C. Hull
  • Published 1 April 2011
  • Mathematics
  • The American Mathematical Monthly
Abstract Margharita P. Beloch was the first person, in 1936, to realize that origami (paperfolding) constructions can solve general cubic equations and thus are more powerful than straightedge and compass constructions. We present her proof. In doing this we use a delightful (and mostly forgotten?) geometric method due to Eduard Lill for finding the real roots of polynomial equations. 
View on Taylor & Francis
mars.wne.edu
31 Citations
Highly Influential Citations
3
Background Citations
14
Methods Citations
5
Results Citations
1

31 Citations

Citation Type

Citation Type

Folding Cubic Roots: Margherita Piazzolla Beloch’s Contribution to Elementary Geometric Constructions
In this paper we describe the contributions made by Margherita Piazzolla Beloch to the field of elementary geometric construction. In particular we examine her solution of the classical Greek problem…
On the Mechanization of Straightedge and Compass Constructions
TLDR
This paper discusses the mechanization of straightedge and compass constructions and focuses on the algebraic approaches and presents two methods which are implemented; one is due to Lebesgue and the other one was jointly designed by Gao and Chou.
From Ancient Greece to Beloch's Crease: The Delian Problem and Origami
The problem of how to double the volume of a cube, also known as the Delian problem, has intrigued mathematicians for millennia. The great variety of solutions discovered over the centuries have used…
Septic equations are solvable by 2-fold origami
In this paper we prove that a generic rational equation of degree $7$ is solvable by 2-fold origami. In particular we show how to septisect an arbitrary angle. This extends the work of Alperin & Lang…
The Parbelos, a Parabolic Analog of the Arbelos
TLDR
A parabolic analog of the arbelos is introduced, using theorems of Archimedes and Lambert to demonstrate seven properties of the parbelos, drawing analogies to similar properties ofThe arbelo, some of which may be new.
Geometric solution of a quintic equation by two-fold origami
This article shows how to find the solution of an arbitrary quintic equation by performing two simultaneous folds on a sheet of paper. The folds achieve specific incidences between a set of point and…
Existence of a Solution for Beloch’s Fold
In her work, Beloch showed that the fold may be applied to solve arbitrary cubic equations. The operation is considered today as one of the so-called “axioms” of origami (the Japanese art of paper…
Form and art of closed crease origami
We show some origami models appeared for the first time as student projects in Bauhaus around 1930. The same models are studied today by mathematicians and structural engineers for its geometric and…
Division of an angle into equal parts and construction of regular polygons by multi-fold origami
This article analyses geometric constructions by origami when up to $n$ simultaneous folds may be done at each step. It shows that any arbitrary angle can be $m$-sected if the largest prime factor of…
Polynomials with Closed Lill Paths
Lill’s method is a striking 19th century technique for discovering real roots of polynomials with real coefficients. It is illustrated for the polynomial p(x) = 4x4 + 6x3 + 5x2 + 4x + 1 in FIGURE 1.…
...
...

References

SHOWING 1-10 OF 27 REFERENCES
The power of multifolds: Folding the algebraic closure of the rational numbers
It is well known that the usual Huzita-Hatori axioms for origami enable angle trisection but not angle quintisection. Using the concept of a multifold, Lang has achieved quintisection but not…
Geometry and Number Theory on Clovers
TLDR
The lemniscate can be divided into n arcs of equal length if and only if n is a power of 2 times a product of distinct Fermat primes, and by an earlier theorem of Gauss these are the values of n for which a regular n-gon is constructible by straightedge and compass.
MORE ON PAPERFOLDING
It is a common knowledge that folding a sheet of paper yields a straight line. We start our discussion of paperfolding with a mathematical explanation of this phenomenon. The model for a paper sheet…
A Mathematical Theory of Origami Constructions and Numbers
In this article we give a simplified set of axioms for mathematical origami and numbers. The axioms are hierarchically structured so that the addition of each axiom, allowing new geometrical…
Famous Problems of Elementary Geometry
OUR mathematical readers who do not read German will be glad to know that they have now before them a translation of a discussion of three famous geometric problems of antiquity, namely, the…
Totally Real Origami and Impossible Paper Folding
TLDR
It is discovered that it is impossible to construct a cube with twice the volume of a given cube using origami, just as it was impossible to do using a compass and straight edge.
One-, Two-, and Multi-Fold Origami Axioms
In 1989 [15], Huzita introduced the six origami operations that have now become know as the Huzita Axioms (HAs). The HAs, shown in Figure 1, constitute six distinct ways of defining a single fold by…
Geometric Solutions to Algebraic Equations
A method to the explict solutions of general systems of algebraic equations is presented via the metric form of affiliated K\"ahler manifolds. The solutions to these systems arise from sets of…
A Note on “Impossible” Paper Folding
TLDR
A Note on “Impossible” Paper Folding is written on behalf of the American Mathematical Monthly.
Origami and Geometric Constructions
1 Origami and Geometric Constructions1 By Robert J. Lang Copyright ©1996–2015. All rights reserved. Introduction…
...
...