Synthetic geometry

Summary

Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulate, and at present called axioms. The term "synthetic geometry" has been coined only after the 17th century, and the introduction by René Descartes of the coordinate method, which was called analytic geometry. So the term "synthetic geometry" was introduced to refer to the older methods that were, before Descartes, the only known ones. According to Felix Klein Synthetic geometry is that which studies figures as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae as can be written down after the adoption of an appropriate system of coordinates. The first systematic approach for synthetic geometry is Euclid's Elements. However, it appeared at the end of the 19th century that Euclid's postulates

Official source

https://en.wikipedia.org/wiki/Synthetic_geometry

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (6)

Related publications

Related people

Related units

Related concepts (49)

Related concepts

Related courses (1)

This course provides an introduction to particle-based methods for the numerical resolution of partial differential equations describing continuum phenomena or for the simulation of particulate flows. Details are given for the Material Point Method (MPM) and the Discrete Element Method (DEM).

Related courses

Related lectures (2)

Related lectures

Theory-Guided Enhancement of CO2 Reduction to Ethanol on Ag-Cu Tandem Catalysts via Particle-Size Effects

Raffaella Buonsanti, Pranit Srinivas Iyengar, James Richard Pankhurst

In the CO2 reduction reaction, the design of electrocatalysts that selectively promote alcohols over hydrocarbons (e.g., ethanol over ethylene) hinges on the understanding of the pathways and specific sites that form alcohols. Herein, theoretical considerations guide state-of-the-art synthesis of well-defined catalysts to show that higher selectivity toward ethanol is achieved on Cu(110) edge sites compared to Cu(100) terraces. Specifically, we study the catalytic behavior of Cu nano-cubes (Cu-cub) of different sizes in the framework of tandem catalysis with CO-producing Ag nano-spheres. We predict and experimentally find that the smaller Cu-cub possess higher selectivity for ethanol in view of their larger edge-to-faces ratio and of the fact that ethylene is produced at terraces while ethanol is selectively produced at step edges. These results call for synthetic developments toward Cu nanostructures exposing only edge sites, such as hollow cubic nanocages, to further increase ethanol selectivity. More generally, this study encourages the application of well-defined nano catalysts as a bridge between theory and experiments in electrocatalysis.

AMER CHEMICAL SOC2021

, , , , ,

Tube drawing with a ﬁxed plug is investigated experimentally and numerically in order to optimise an industrial pass sequence. Tube drawings are performed on a laboratory drawing bench equipped with force, displacement and temperature sensors. The drawing limit of 316 LVM stainless steel (SS) tubes is determined by varying the plug position and the eﬀect of lubricant is discussed. The numerical model relies on the precise die geometry measured using a contact method and on the mechanical behaviour of SS 316 LVM characterised by tensile tests between 20 °C and 200 °C on annealed tubes. The friction coeﬃcients between tube and die and tube and plug are determined by FE modelling by comparing measured and simulated plug, die and drawing forces. The use of a solid lubricant together with the lubricating oil prevents stick and slip and lowers the friction coeﬃcients. Friction is further decreased by preheating the die. The identiﬁed friction coeﬃcients are discussed in terms of the classic Stribeck friction curves and lubrication is found to be in the hydrodynamic regime.

2018

Insights into Reaction Intermediates to Predict Synthetic Pathways for Shape-Controlled Metal Nanocrystals

Raffaella Buonsanti, Shubhajit Das, Pranit Srinivas Iyengar, Anna Loiudice, Valeria Mantella, James Richard Pankhurst, Michal Stanislaw Strach

Understanding nucleation phenomena is crucial across all branches of physical and natural sciences. Colloidal nanocrystals are among the most versatile and tunable synthetic nanomaterials. While huge steps have been made in their synthetic development, synthesis by design is still impeded by the lack of knowledge of reaction mechanisms. Here, we report on the investigation of the reaction intermediates in high temperature syntheses of copper nanocrystals by a variety of techniques, including X-ray absorption at a synchrotron source using a customized in situ cell. We reveal unique insights into the chemical nature of the reaction intermediates and into their role in determining the final shape of the metal nanocrystals. Overall, this study highlights the importance of understanding the chemistry behind nucleation as a key parameter to predict synthetic pathways for shape-controlled nanocrystals.

AMER CHEMICAL SOC2019