# Poncelet’s porism: a long story of renewed discoveries, I

@article{Centina2016PonceletsPA, title={Poncelet’s porism: a long story of renewed discoveries, I}, author={Andrea Del Centina}, journal={Archive for History of Exact Sciences}, year={2016}, volume={70}, pages={1-122} }

In 1813, J.-V. Poncelet discovered that if there exists a polygon of n-sides, which is inscribed in a given conic and circumscribed about another conic, then infinitely many such polygons exist. This theorem became known as Poncelet’s porism, and the related polygons were called Poncelet’s polygons. In this article, we trace the history of the research about the existence of such polygons, from the “prehistorical” work of W. Chapple, of the middle of the eighteenth century, to the modern… Expand

#### Figures from this paper

figure 1 figure 2 figure 3 figure 4 figure 5 figure 6 figure 7 figure 8 figure 9 figure 10 figure 11 figure 12 figure 13 figure 14 figure 15 figure 16 figure 17 figure 18 figure 19 figure 20 figure 21 figure 22 figure 23 figure 24 figure 25 figure 26 figure 27 figure 28 figure 29 figure 30 figure 31 figure 32 figure 33 figure 34 figure 35 figure 36

#### 13 Citations

Pascal’s mystic hexagram, and a conjectural restoration of his lost treatise on conic sections

- Philosophy
- 2020

Through an in-depth analysis of the notes that Leibniz made while reading Pascal’s manuscript treatise on conic sections, we aim to show the real extension of what he called “hexagrammum mysticum”,… Expand

The recognition and the constitution of the theorems of closure

- Philosophy
- 2018

Abstract This papers analyzes how several geometric theorems, that were considered to be disconnected from each other in the beginning of the nineteenth century, have been progressively recognized as… Expand

Boscovich's geometrical principle of continuity, and the “mysteries of the infinity”

- Philosophy
- 2018

Abstract In this paper we give a detailed account of Boscovich's geometrical principle of continuity. We also compare his ideas with those of his forerunners and successors, in order to cast some… Expand

Carnot’s theory of transversals and its applications by Servois and Brianchon: the awakening of synthetic geometry in France

- Mathematics
- 2021

In this paper we discuss in some depth the main theorems pertaining to Carnot’s theory of transversals, their initial reception by Servois, and the applications that Brianchon made of them to the… Expand

Poncelet plectra: harmonious curves in cosine space

- Mathematics, Computer Science
- Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
- 2021

It has been shown that the family of Poncelet N-gons in the confocal pair (elliptic billiard) conserves the sum of cosines of its internal angles, and that when N=3, the cosine triples of both families sweep the same planar curve: an equilateral cubic resembling a plectrum (guitar pick). Expand

Poncelet-Darboux, Kippenhahn, and Szeg\H{o}: interactions between projective geometry, matrices and orthogonal polynomials

- Mathematics
- 2021

We study algebraic curves that are envelopes of families of polygons supported on the unit circle T. We address, in particular, a characterization of such curves of minimal class and show that all… Expand

Closed chains of conics carrying poncelet triangles

- Mathematics
- 2017

We investigate closed chains of conics which carry Poncelet triangles. In particular, we show that every chain of conics which carries Poncelet triangles can be closed. Furthermore, for $$k=3$$k=3… Expand

Loci of Poncelet Triangles with Multiple Caustics

- Computer Science, Mathematics
- ArXiv
- 2021

It is shown that despite a more complicated dynamic geometry, the locus of certain triangle centers and associated points remain conics and/or circles. Expand

An App for the Discovery of Properties of Poncelet Triangles

- Computer Science, Mathematics
- 2021

We describe a newly-developed, free, browser-based application, for the interactive exploration of the dynamic geometry of Poncelet families of triangles. The main focus is on responsive display of… Expand

On two extensions of Poncelet theorem

- Mathematics
- 2020

Abstract In this note we provide two extensions of a particular case of Poncelet theorem.

#### References

SHOWING 1-10 OF 121 REFERENCES

I. On the porism of the in-and-circumscribed polygon

- Mathematics
- Proceedings of the Royal Society of London
- 1862

The Porism referred to is as follows, viz. two conics may be so related to each other, that a polygon may be inscribed in the one, and circumscribed about the other conic, in such manner that any… Expand

Mathematical Thought from Ancient to Modern Times

- Mathematics
- 1972

This comprehensive history traces the development of mathematical ideas and the careers of the men responsible for them. Volume 1 looks at the discipline's origins in Babylon and Egypt, the creation… Expand

Hidden Harmony―Geometric Fantasies: The Rise of Complex Function Theory

- Mathematics
- 2013

The book is a history of complex function theory from its origins to 1914, when the essential features of the modern theory were in place. It is the first history of mathematics devoted to complex… Expand

James Joseph Sylvester: Jewish Mathematician in a Victorian World

- 1998

Here, in this first biographical study of James Joseph Sylvester, Karen Hunger Parshall makes a signal contribution to the history of mathematics, Victorian history, and the history of science. A… Expand

Toward a History of Nineteenth-Century Invariant Theory

- Mathematics
- 1989

Publisher Summary Before Gauss considered binary forms in his Disquisitiones Arithmeticae , Joseph-Louis Lagrange had encountered and dealt with the problem of transformation of homogeneous… Expand

Bicentennial of the Great Poncelet Theorem (1813-2013): Current Advances

- Mathematics, Physics
- 2012

The paper gives a review of very recent results related to the Poncelet Theorem, on the occasion of its bicentennial. We are telling the story of one of the most beautiful theorems of Geometry,… Expand

The British development of the theory of invariants (1841–1895)

- Mathematics
- 2006

The two main British exponents of the theory of invariants, Arthur Cayley and James Joseph Sylvester, first encountered the idea of an “invariant” in an 1841 paper by George Boole. In the 1850s,… Expand

Poncelet theorems

- Mathematics
- 1995

The aim of this note is to collect some more or less classical theorems of Poncelet type and to provide them with short modern proofs. Where classical geometers used elliptic functions (or angular… Expand

The rise of Cayley's invariant theory (1841–1862)

- Mathematics
- 1986

Abstract In his pioneering papers of 1845 and 1846, Arthur Cayley (1821–1895) introduced several approaches to invariant theory, the most prominent being the method of hyperdeterminant derivation.… Expand