The three classic problems of Greek mathematics are the squaring of the circle, the trisection of the angle and the doubling of the cube. These questions had enormous importance in the progress of Hellenic geometry and acted as true poles of interest, guiding the research of the great mathematicians of Antiquity. This article analyzes in its historical context the genesis of the cissoid, a curve specifically conceived by Diocles to provide a new solution to the problem of doubling the cube, and its influence on the development of Greek mathematics is studied.

It is currently unknown if surgeons and biomaterial scientists &or tissue engineers (BS&orTE) process and evaluate information in similar or different (un)biased ways. For the gold standard of surgery to move "from bench to bedside," there must naturally be synergies between these key stakeholders' perspectives. Because only a small number of biomaterials and tissue engineering innovations have been translated into the clinic today, we hypothesized that this lack of translation is rooted in the psychology of surgeons and BS&orTE. Presently, both clinicians and researchers doubt the compatibility of surgery and research in their daily routines. This has led to the use of a metaphorical expression "squaring of the circle," which implies an unsolvable challenge. As bone tissue engineering belongs to the top five research areas in tissue engineering, we choose the field of bone defect treatment options for our bias study. Our study uses an online survey instrument for data capture such as incorporating a behavioral economics cognitive framing experiment methodology. Our study sample consisted of surgeons (n = 208) and BS&orTE (n = 59). And we used a convenience sampling method, with participants (conference attendants) being approached both in person and through email between October 22, 2022, and March 13, 2023. We find no distinct positive-negative cognitive framing differences by occupation. That is, any framing bias present in this surgical decision-making setting does not appear to differ significantly between surgeon and BS&orTE specialization. When we explored within-group differences by frames, we see statistically significant (p < 0.05) results for surgeons in the positive frame ranking autologous bone graft transplantation lower than surgeons in the negative frame. Furthermore, surgeons in the positive frame rank Ilizarov bone transport method higher than surgeons in the negative frame (p < 0.05).

I dedicate this article, in Memoriam, to Professor Sylwester Dworacki, my first guide in Greek texts, with whom I later had the distinguished privilege to frequently discuss diverse issues in philological exegesis. The little-known figure of Hippocrates of Chios has recently attracted strong interest of several scholars, though mainly by historians of mathematics. Aristotle mentioned critically his quadrature of the circle by means of segments or by means of lunules. Aristotle’s commentator Simplicius, citing Eudemus of Rhodos, quoted a longer paraphrase of Hippocrates’ arguments regarding the quadrature of the lunules. Appropriately selected parts from these arguments are given here in Greek, along with their faithful Polish translation. One should carefully understand the critical stance of Aristotle, who in his particular way understood quadrature as the finding of the geometrical mean and, therefore, accused Hippocrates of using false diagrams.

We prove a measurable version of the Hall marriage theorem for actions of finitely generated abelian groups. In particular, it implies that for free measure-preserving actions of such groups and measurable sets which are suitably equidistributed with respect to the action, if they are are equidecomposable, then they are equidecomposable using measurable pieces. The latter generalizes a recent result of Grabowski, Máthé and Pikhurko on the measurable circle squaring and confirms a special case of a conjecture of Gardner.

Squaring the circle today

The Squaring of the Circle:

If you pose the question given in the title of this note you will listen a negative answer. In the Google searcher you will receive about 6 million results. It means to find something new in the problem formulated by Greek mathematician is useless. This problem alongside with the circle squaring is considered as undecidable problem.

Squaring the Circle in a Mirror

Trying to Square the Circle: Comparative Remarks on the Rights of the Surviving Spouse on Intestacy

103.13 Squaring the circle by using proportion

When the mobility of students is being discussed, it is often compared to the squaring of the circle – hence an insoluble task. A conference of the ombudsmans service for students with the title „Anerkennungen – Durchlassigkeit studienrechliche Gegensatze! Wie behandeln?” (Recognition – Permeability of Contrasts of Study Matter. How to handle?) was dedicated to this topic in November 2018. The article represents a slightly reviewed and extended version of the lecture which was held by the author at this conference.

This paper focuses on the mathematisation of mechanics in the seventeenth century, specifically on how the representation of compounded rectilinear motions presented in the ancient Greek Mechanica found its way into Newton’s Principia almost two thousand years later. I aim to show that the path from the former to the latter was optical: the conceptualisation of geometrical lines as paths of reflection created a physical interpretation of dia­grammatic principles of geometrical point-motion, involving the kinematics and dynamics of light reflection. Upon the atomistic conception of light, the optical interpretation of such geometrical principles entailed their mechanical generalisation to local motion; rectilinear motion via the physico-mathemat­ics of reflection and the Mechanica’s parallelogram rule; circular motion via the physico-mathematics of reflection, the Archimedean squaring of the circle and the Mechanica’s extension of the parallelogram rule to centripetal motion. This appeal to the physico-mathematics of reflection forged a realist founda­tion for the mathematisation of motion. Whereas Aristotle’s physics rested on motions which had their source in the nature of the elements, early modern thinkers such as Harriot, Descartes, and Newton based their new principles of mechanical motion upon selected elements of the mechanics of light motion, projected upon the geometry of the parallelogram rule for rectilinear and, ultimately, circular motion.

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Translation of Menus: Labour of Sisyphos, Squaring the Circle or Marrying Water and Fire?

Article Commentary: Squaring the Circle

Laczkovich proved that if bounded subsets $A$ and $B$ of $R^k$ have the same non-zero Lebesgue measure and the box dimension of the boundary of each set is less than $k$, then there is a partition of $A$ into finitely many parts that can be translated to form a partition of $B$. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-only versions of Tarski's circle squaring and Hilbert's third problem.

From the Museum: the Art of Thinking. Squaring the circle.

방원도의 투영

This article offers some reflections on sustainability in Kolkata, Indian city characterized by a steady increase in population, poverty and violence. It is argued that the key issue is not the continuous increase in population, but how to square the circle between, on the one hand, social justice and the right at a decent standard of living and, on the other, environmental sustainability and the availability of resources. Questo articolo si occupa della questione di sostenibilita in Kolkata, citta indiana caratterizzata da un costante aumento di popolazione, violenza e poverta. Si sostiene che la questione fondamentale non e l’aumento continuo della popolazione, ma come far quadrare il cerchio tra giustizia sociale e diritto a un livello di vita decente, da un lato e, dall’altro la sostenibilita ambientale e la disponibilita di risorse.

Squaring of circle is an unsolved problem with the official  value 3.1415926... with the new  value 1/4 (142 ) it is done in this paper.