Morphogenesis of Curved Structural Envelopes Under Fabrication Constraints
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Descartes, Euler, Poincaré, Pólya and Polyhedra Séminaire De Philosophie Et Mathématiques, 1982, Fascicule 8 « Descartes, Euler, Poincaré, Polya and Polyhedra », , P
Séminaire de philosophie et mathématiques PETER HILTON JEAN PEDERSEN Descartes, Euler, Poincaré, Pólya and Polyhedra Séminaire de Philosophie et Mathématiques, 1982, fascicule 8 « Descartes, Euler, Poincaré, Polya and Polyhedra », , p. 1-17 <http://www.numdam.org/item?id=SPHM_1982___8_A1_0> © École normale supérieure – IREM Paris Nord – École centrale des arts et manufactures, 1982, tous droits réservés. L’accès aux archives de la série « Séminaire de philosophie et mathématiques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ - 1 - DESCARTES, EULER, POINCARÉ, PÓLYA—AND POLYHEDRA by Peter H i l t o n and Jean P e d e rs e n 1. Introduction When geometers talk of polyhedra, they restrict themselves to configurations, made up of vertices. edqes and faces, embedded in three-dimensional Euclidean space. Indeed. their polyhedra are always homeomorphic to thè two- dimensional sphere S1. Here we adopt thè topologists* terminology, wherein dimension is a topological invariant, intrinsic to thè configuration. and not a property of thè ambient space in which thè configuration is located. Thus S 2 is thè surface of thè 3-dimensional ball; and so we find. among thè geometers' polyhedra. thè five Platonic “solids”, together with many other examples. However. we should emphasize that we do not here think of a Platonic “solid” as a solid : we have in mind thè bounding surface of thè solid. -
Polyhedral Surfaces, Discrete Curvatures, Shape Analysis, 3D Modelling, Segmentation
JOURNAL OF MEDICAL INFORMATICS & TECHNOLOGIES Vol. 9/2005, ISSN 1642-6037 polyhedral surfaces, discrete curvatures, shape analysis, 3D modelling, segmentation. Alexandra BAC, Marc DANIEL, Jean-Louis MALTRET* 3D MODELLING AND SEGMENTATION WITH DISCRETE CURVATURES Recent concepts of discrete curvatures are very important for Medical and Computer Aided Geometric Design applications. A first reason is the opportunity to handle a discretisation of a continuous object, with a free choice of the discretisation. A second and most important reason is the possibility to define second-order estimators for discrete objects in order to estimate local shapes and manipulate discrete objects. There is an increasing need to handle polyhedral objects and clouds of points for which only a discrete approach makes sense. These sets of points, once structured (in general meshed with simplexes for surfaces or volumes), can be analysed using these second-order estimators. After a general presentation of the problem, a first approach based on angular defect, is studied. Then, a local approximation approach (mostly by quadrics) is presented. Different possible applications of these techniques are suggested, including the analysis of 2D or 3D images, decimation, segmentation... We finally emphasise different artefacts encountered in the discrete case. 1. INTRODUCTION This paper offers a review of work on discrete curvatures for Computer Aided Geometric Design applications and a discussion about some ideas for further studies. We consider the discrete curvatures computed on a set of points and their relevance to continuous curvatures when the density of the points increases. First of all, it is interesting to recall that curvatures are fundamental tools for studying curves and surfaces. -
Binding Regular Polyhedra
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of the Academy's Library Optimal packages: binding regular polyhedra F. Kovacs´ Department of Structural Mechanics Budapest University of Technology and Economics ABSTRACT: Strings on the surface of gift boxes can be modelled as a special kind of cable-and-joint struc- ture. This paper deals with systems composed of idealised (frictionless) closed loops of strings that provide stable binding to the underlying convex polyhedron (‘package’). Optima are searched in both the sense of topology and geometry in finding minimal number of closed loops as well as the minimal (total) length of cables to ensure such a stable binding for simple cases of polyhedra. 1 INTRODUCTION and concave otherwise (for example, ‘zigzag’ loops with alternating turns are concave). In these terms, the 1.1 Loops on a polyhedral surface loop on the left hand side of Fig. 1 is complex due to the self-intersection at the top and concave because of There are some practical occurrences of closed strings the two rectangular turns with different handedness on polyhedral surfaces. For example, mailed pack- at the bottom (such connections will be called self- ages are traditionally bound by some pieces of string. binding from now on); whereas the skew string to the Another example is from basketry, when a woven right is simple and convex (by having no turns within pattern over a polyhedral surface is also formed by the polyhedral surface at all). some (mainly closed) strings (Tarnai, Kov´acs, Fowler, & Guest 2012, Kov´acs 2014). -
Parity Check Schedules for Hyperbolic Surface Codes
Parity Check Schedules for Hyperbolic Surface Codes Jonathan Conrad September 18, 2017 Bachelor thesis under supervision of Prof. Dr. Barbara M. Terhal The present work was submitted to the Institute for Quantum Information Faculty of Mathematics, Computer Science und Natural Sciences - Faculty 1 First reviewer Second reviewer Prof. Dr. B. M. Terhal Prof. Dr. F. Hassler Acknowledgements Foremost, I would like to express my gratitude to Prof. Dr. Barbara M. Terhal for giving me the opportunity to complete this thesis under her guidance, insightful discussions and her helpful advice. I would like to give my sincere thanks to Dr. Nikolas P. Breuckmann for his patience with all my questions and his good advice. Further, I would also like to thank Dr. Kasper Duivenvoorden and Christophe Vuillot for many interesting and helpful discussions. In general, I would like to thank the IQI for its hospitality and everyone in it for making my stay a very joyful experience. As always, I am grateful for the constant support of my family and friends. Table of Contents 1 Introduction 1 2 Preliminaries 2 3 Stabilizer Codes 5 3.1 Errors . 6 3.2 TheBitFlipCode ............................... 6 3.3 The Toric Code . 11 4 Homological Code Construction 15 4.1 Z2-Homology .................................. 18 5 Hyperbolic Surface Codes 21 5.1 The Gauss-Bonnet Theorem . 21 5.2 The Small Stellated Dodecahedron as a Hyperbolic Surface Code . 26 6 Parity Check Scheduling 32 6.1 Seperate Scheduling . 33 6.2 Interleaved Scheduling . 39 6.3 Coloring the Small Stellated Dodecahedron . 45 7 Fault Tolerant Quantum Computation 47 7.1 Fault Tolerance . -
Winter Break "Homework Zero"
\HOMEWORK 0" Last updated Friday 27th December, 2019 at 2:44am. (Not officially to be turned in) Try your best for the following. Using the first half of Shurman's text is a good start. I will possibly add some notes to supplement this. There are also some geometric puzzles below for you to think about. Suggested winter break homework - Learn the proof to Cauchy-Schwarz inequality. - Find out what open sets, closed sets, compact set (closed and bounded in Rn, as well as the definition with open covering cf. Heine-Borel theorem), and connected set means in Rn. - Find out what supremum (least upper bound) and infimum (greatest lower bound) of a subset of real numbers mean. - Find out what it means for a sequence and series to converge. - Find out what does it mean for a function f : Rn ! Rm to be continuous. There are three equivalent characterizations: (1) Limit preserving definition. (2) Inverse image of open sets are open. (3) −δ definition. - Find the statements to extreme value theorem, intermediate value theorem, and mean value theorem. - Find out what is the fundamental theorem of calculus. - Find out how to perform matrix multiplication, computation of matrix inverse, and matrix deter- minant. - Find out what is the geometric meaning of matrix determinantDecember, 2019 . - Find out how to solve a linear system of equation using row reduction. n m - Find out what a linear map from R to R is, and what is its relationth to matrix multiplication by an m × n matrix. How is matrix product related to composition of linear maps? - Find out what it means to be differentiable at a point a for a multivariate function, what is the derivative of such function at a, and how to compute it. -
Elysée Wilson-Egolf Topic: Polygons, Polyhedra, Polytope Series
Berkeley Math Circle Intermediate I, 1/23, 1/20, 2/6 Presenter: Elysée Wilson-Egolf Topic: Polygons, Polyhedra, Polytope Series Part 1 – Polygon Angle Formula Let’s start simple. How do we find the sum of the interior angles of a convex polygon? • Write down as many methods as you can. • Think about: how can you adapt your method to concave polygons? Formula for the interior angles of a polygon: _________________ . Q: There is an equality between the number of ways to triangulate an n+2-gon and the number of expressions containing n pairs of parentheses that are correctly matched. Prove that these numbers are the same. Can you find a closed formula? Part 2 – Platonic Solids and Angular Defect Definition: A platonic solid is a polyhedron which a. has only one kind of regular polygon for a face and b. has the same number of faces at each vertex. One simple example is that of a cube. Each face is a square (=regular quadrilateral) and each vertex is connected to exactly three squares. In Schläfli notation, this is {4, 3}: the regular polyhedron with 3 regular 4-gons at each vertex. • Can there be polyhedra with exactly one or two squares at each vertex? • Can there be polyhedra with exactly four squares at each vertex? For future reference, we’ll need the notion of the “angular defect” as well. Rather than focusing on what is present at the vertex, we focus on what is absent: the deviation from 360 degrees at the vertex, or “leftover” angle. In this case, it is 360−90×3=90 . -
BBC 4 Listings for 15 – 21 April 2017 Page 1 of 5 SATURDAY 15 APRIL 2017 UB40
BBC 4 Listings for 15 – 21 April 2017 Page 1 of 5 SATURDAY 15 APRIL 2017 UB40. hospital and heralded a golden age of philanthropy. SAT 19:00 The Silk Road (p03qb3q4) Exploring historical documents and artefacts, Amanda Episode 3 SAT 00:05 Chas & Dave: Last Orders (b01nkdsv) examines the plight of women in Georgian London, particularly Documentary which highlights cockney duo Chas & Dave's how the attitudes of the time led mothers to abandon their In the final episode of his series tracing the story of the most rich, unsung pedigree in the music world and a career spanning babies at the hospital. Tom looks at the momentous trials and famous trade route in history, Dr Sam Willis continues his 50 years, almost the entire history of UK pop. They played with tribulations faced by Handel in London and discovers how the journey west in Iran. The first BBC documentary team to be everyone from Jerry Lee Lewis to Gene Vincent, toured with composer became involved with the Foundling Hospital granted entry for nearly a decade, Sam begins in the legendary The Beatles, opened for Led Zeppelin at Knebworth - and yet alongside another philanthropist of the day, the artist William city of Persepolis - heart of the first Persian Empire. are known mainly just for their cheery singalongs and novelty Hogarth. records about snooker and Spurs. Following an ancient caravan route through Persia's deserts, he visits a Zoroastrian temple where a holy fire has burned for The film also looks at the pair's place among the great musical SUN 20:00 Darcey Bussell's Looking for Audrey 1,500 years, and Esfahan, one of the Silk Road's architectural commentators on London life - and in particular the influence (b04w7mfk) jewels and rival to Sam's next destination - Istanbul. -
2012 Annual Report
AnnuAl RepoRt 2012 On the cover: Brighamia insignis, commonly known as ‘ālula or ‘ōlulu in Hawaiian, a critically endangered plant endemic to Kaua‘i. This Page: Bamboo Grove, Allerton Garden, Kaua‘i Message froM Chipper WiChMan and Merrill MagoWan 2012 was an important year for the National Tropical Botanical Garden in many ways. One of the most significant was the fact that it marked the first year of our new five-year strategic plan. This plan is our roadmap to achieving our vision and our potential as a leading botanic institution. The plan represents our dreams and aspirations for the future and the first year demonstrated great progress towards the challenging goals we set for ourselves. Two significant key goals of the plan call for the creation of an international center for tropical botany at The Kampong (our garden in Florida) in collaboration with Florida International University and the renewal and improvement of our flagship garden – McBryde Garden. Both of these goals will extend the impact of our organization to a national and international audience as well as help to create a more sustainable organization financially. Significant contributions were received in 2012 towards both of these goals. Another highlight of 2012 was the fall Board meeting held in the United Kingdom. In the 49-year history of our organization, this is the first time the Board has met outside of the United States. The meeting took us to the Eden Project in Cornwall and the Royal Botanic Garden Edinburgh in Scotland where they shared their expertise in innovation through marketing, visitor services and education. -
Chapter 2 Figures and Shapes 2.1 Polyhedron in N-Dimension in Linear
Chapter 2 Figures and Shapes 2.1 Polyhedron in n-dimension In linear programming we know about the simplex method which is so named because the feasible region can be decomposed into simplexes. A zero-dimensional simplex is a point, an 1D simplex is a straight line segment, a 2D simplex is a triangle, a 3D simplex is a tetrahedron. In general, a n-dimensional simplex has n+1 vertices not all contained in a (n-1)- dimensional hyperplane. Therefore simplex is the simplest building block in the space it belongs. An n-dimensional polyhedron can be constructed from simplexes with only possible common face as their intersections. Such a definition is a bit vague and such a figure need not be connected. Connected polyhedron is the prototype of a closed manifold. We may use vertices, edges and faces (hyperplanes) to define a polyhedron. A polyhedron is convex if all convex linear combinations of the vertices Vi are inside itself, i.e. i Vi is contained inside for all i 0 and all _ i i 1. i If a polyhedron is not convex, then the smallest convex set which contains it is called the convex hull of the polyhedron. Separating hyperplane Theorem For any given point outside a convex set, there exists a hyperplane with this given point on one side of it and the entire convex set on the other. Proof: Because the given point will be outside one of the supporting hyperplanes of the convex set. 2.2 Platonic Solids Known to Plato (about 500 B.C.) and explained in the Elements (Book XIII) of Euclid (about 300 B.C.), these solids are governed by the rules that the faces are the regular polygons of a single species and the corners (vertices) are all alike. -
Eden Geothermal Limited the Eden Project Bodelva Par PL24 2SG Tel
Eden Geothermal Limited The Eden Project Bodelva Par PL24 2SG Tel: +44 (0)1726 806545 E: [email protected] Date: 18th January 2021 INVITATION TO TENDER Dear Sir/Madam Project Eden Geothermal Project Tender Name Geology services during deep geothermal development Tender reference EGL-ITT-C088 You are invited to submit a competitive tender for geology services for a project co-funded by the European Regional Development Fund. Please submit your proposal in full no later than: Wednesday 10th February 2021 at 16:00 hours Except under exceptional circumstances, no extension of time and date by which the tender must be submitted will be granted. Late submissions will not be accepted. Any query in connection with this tender or the invitation to tender shall be submitted in writing (by email) to [email protected] by: Monday 1st February 2021 at 12.00 noon We look forward to receiving your submission. Yours faithfully Augusta Grand Executive Director EGL-ITT-C088 Ver3.0 Page 1 of 80 This page intentionally blank EGL-ITT-C088 Ver3.0 Page 2 of 80 Invitation to Tender: Geology services during deep geothermal development Project Eden Geothermal Project Tender reference EGL-ITT-C088 Revision Ver 3.0 Release Date 18th January 2021 Issuer Eden Geothermal Limited (“EGL”) Supplier Response Date 10th February 2021 at 16.00 EGL-ITT-C088 Ver3.0 Page 3 of 80 Contents PART A: INSTRUCTIONS TO TENDERERS 1 Instructions for Completion 7 1.1 Submission Details 7 1.2 Enquiries and Tender Queries 8 1.3 Format of Tender Submission 8 1.4 Project Description -
Oak Tree Cottage, Sandplace, Looe, Cornwall Pl13 1Pj Guide Price £725,000
OAK TREE COTTAGE, SANDPLACE, LOOE, CORNWALL PL13 1PJ GUIDE PRICE £725,000 DULOE 2 MILES, LOOE 2 MILES, LISKEARD 7 MILES, PLYMOUTH 20 MILES, EXETER 61 MILES THE UNSPOILT EAST LOOE RIVER VALLEY - Oozing with warmth and character, the quintessential Cornish cottage, immaculately presented and successfully blending traditional, contemporary and bespoke features, set within beautiful and private landscaped gardens. About 2273 sq ft, Reception Hall, 13' Kitchen/Breakfast Room, 18' Dining Room, 22' Triple Aspect Sitting Room with wood burner, Study, 19' Luxury Master Bedroom with Private Shower/WC, 3 Further Bedrooms (1 Ensuite), Family Bathroom, Level Drive/Parking, Double Garage, Studio, Various Outbuildings. LOCATION Oak Tree Cottage lies in the sheltered, unspoilt and wooded countryside of the valley of the East Looe River and enjoys a south westerly aspect over the sylvan landscape. This unspoilt landscape is designated as an Area of Great Landscape Value and the property lies adjacent to a County Wildlife Site. The scattered rural hamlet of Sandplace is a short distance north of the South Cornish Coast and beaches at Looe. the popular rural village of Duloe with its community shop, award winning local inn, primary school (rated "good" by Ofsted) and places of worship. A regular bus service provides convenient connections with Liskeard, Looe and Polperro. The Cornwall Area of Outstanding Natural Beauty lies a short distance to the south with its unspoilt coastal scenery, hidden coves and creeks, with various areas of the coast in the ownership of The National Trust. Nearby, Polperro, Polruan/Fowey and Looe are picturesque, each with their own harbours and fishing fleets. -
Are Proud to Support the Golowan Festival • Litho Printing • Business Stationery
1 Digital Peninsula Network Cornwall’s leading provider of training and apprenticeships in: Internet Marketing, Web Design, Social Media Marketing, Graphic Design, Video Production and much, much more. t: 01736 333700 e: [email protected] For further information about our training courses w: www.digitalpeninsula.com please contact our office on 01736 333700. Digital Peninsula Network Limited 1 & 2 Old Brewery Yard Wherever possible DPN will access funding to Penzance, Cornwall, TR18 2SL subsidise training costs. Please ask for more details. 2 Photograph by John Stedman Golowan – All At Sea Welcome to the Golowan Festival – the 27th since its revival. In keeping with Penzance’s wonderful location our theme this year is “Golowan – All at Sea” - a tongue in cheek invitation to celebrate our amazing maritime heritage, and reflect it in a blaze of nautical imagery. However, “All at Sea!” couldn’t be wider of the mark, as the focus of everyone has been to ensure that, while cherishing established traditions, new things are tried, new partnerships secured and Golowan goes from strength to strength. All the elements for a great weekend are here - the mid-summer torchlit procession, serpent dancing and fireworks, the manic Mock Mayor of the Quay ceremony, music and market stalls, parades… and so much more… Golowan’s Quay Fair offers all the fun of the fair on Battery Road from 21st – 25th June and – last, but not least – Golowan 2017 sees the return of the marquee in St Anthony’s Car Park where Big Tow Productions will host a range of events – including the Mock Mayor election.