Geometry And Topology

This paper examines two features of Kant’s 1768 critique of Leibniz’ conception of space. Firstly, Leibniz’ proposed geometrical calculus called ‘analysis situs’; secondly, Leibniz’ relational conception of space. The main thesis of the paper is that Kant’s arguments are more powerful than generally recognized. With regard to the analysis situs, I will show that Kant was quite well informed about this proposed science and that his arguments severely undermine Leibniz’ claims to what it could perform. With regard to the relational theory of space, Kant’s argument would require Leibniz to present a complex story about the relation between God’s act of creation and our spatial experience to defend his relational view, rather than using the simple principle of sufficient reason.

A more general class than complete intersection singularities is the class of determinantal singularities. They are defined by the vanishing of all the minors of a certain size of a m × n-matrix. In this note, we consider G-finite determinacy of matrices defining a special class of determinantal varieties. They are called essentially isolated determinantal singularities (EIDS) and were defined by Ebeling and Gusein-Zade [7]. In this note, we prove that matrices parametrized by generic homogeneous forms of degree d define EIDS. It follows that G-finite determinacy of matrices hold in general. As a consequence, EIDS of a given type (m, n, t) holds in general.

This paper synergizes the newly discovered geometry of all Metallic Means and the recently published mathematical formulae those provide the precise correlations between different Metallic Ratios. The paper also illustrates the concept of the "TRIADS of Metallic Means". The Metallic Means and their TRIADS can be geometrically substantiated, in an intriguing manner, as described herein.

The many-worlds interpretation (MWI) implies that all possible outcomes of quantum measurements are physically realized in some "world" or universe.A topological construct can be developed, and used as a framework in order to view the quantum branching 4-dimensionally. Space undergoes geometrical expansion as it advances from one tд to the next - we can view this expansion through a Hypersphere, Tesseract, or Manifold.  The tд construct views the forward propagation of time as as a function of the total change of energy with respect to the topological phase space it occupies.

All the important parameters of a truncated dodecahedron (having 20 congruent equilateral triangular & 12 congruent regular decagonal faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedron. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of regular n-polyhedrons.

A planetary trains for multi-speed is mainly used for automation in industries of automobile. A planetary
gear train is represented by a graph. It is identified by (i) number of vertices and their connectivity (ii) number of edges and their types and values (iii) fundamental circuits, their size and adjancy. Connectivity of individual link is a property characteristic of kinematic chain. It is possible to identify a planetary gear,therefore of using sets of labele (decimal numbers representing connectivity ) of individual link. The
connectivity of vertices , edges values and circuit values, related to design invariants which in turn indicates the possible behavior of the gear train ( for example capacity of power transmission, speed ratio and power carculation). For a specified degree – of – freedom a number of planetary gear kinematic chain (PGKCs) are selected and hence planetary gear trains (PGTs) can be formed with a given number of links
and joints so that designer must be able to select to select the best train from the view point of say velocity ratio and capacity of power transmission, space requirements etc. Synthesis of planetary gear kinematic chain and planetary gear trains has been studied(1-9). Almost all reported work deals with only
identification of distinct chains. Besides providing an atlus of chains, this in itself does not provide any help to designer in the selection of best possible gear train. In the present paper a simple method based of circuit property ( based on link-link shortest path distance and degree of links) is presented to determine the topology values of power transmission efficiency and topology power transmission capacity of five-links PGKCs and their distinct inversions

A century and a half ago, a revolution in human thought began that has gone largely unrecognized by modern scholars: A system of non-Euclidean geometries was developed that literally changed the way that we view our world. At first, some thought that space itself was non-Euclidean and four-dimensional, but Einstein ended their 'speculations' when he declared that time was the fourth dimension. Yet our commonly perceived space is truly four-dimensional. Einstein unwittingly circumvented that particular revolution in thought and delayed its completion for a later day, although his work was also necessary for the completion of that revolution. That later day is now approaching. The natural progress of science has brought us back to the point where science again needs to consider the physical reality of a higher-dimensional space. Science must acknowledge the truth that space is four-dimensional and space-time is five-dimensional, as required by accepted physical theories and observations, before it can move forward with a new unified fundamental theory of physical reality.

This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Gödel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have a place of pride. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, etc. Perhaps, by Gödel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find the solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Gödel's incompleteness theorems will have ramifications in other areas involving logic. This paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics. [Published in international mathematics journal. Acknowledgments: The author expresses his gratitude to the referees and the Editor-in-Chief for their valuable comments in strengthening the contents of this paper.]

This text is intended for a brief introductory course in plane geometry. It covers the topics from elementary geometry that are most likely to be required for more advanced mathematics courses. The only prerequisite is a semester of algebra. The emphasis is on applying basic geometric principles to the numerical solution of problems. For this purpose the number of theorems and definitions is kept small. Proofs are short and intuitive, mostly in the style of those found in a typical trigonometry or precalculus text. There is little attempt to teach theorem proving or formal methods of reasoning. However the topics are ordered so that they may be taught deductively. The problems are arranged in pairs so that just the odd-numbered or just the even-numbered can be assigned. For assistance, the student may refer to a large number of completely worked-out examples. Most problems are presented in diagram form so that the difficulty of translating words into pictures is avoided. Many proble...

Contains Julian Roberts's original 2002 Subtraction Cutting online 'school lessons':
The 'Absolute Basics of Geometry', 'Tunnel Technique', 'Plug Technique', and 'Displacement Technique';
plus Julian's 2006 Professorial Lecture notes 'Against the Grain: Adventures in Creative Pattern Cutting';
his 2010 Design Manifesto 'Pattern Landscapes';
and an 'idiots guide' to making a fabric tube in preparation for the 'Tunnel Technique', involving Julian's two cats Treacle and Lulu.

Unlike nineteenth-century Gothic fiction, which tends to fixate on the past, the haunted and the ghostly, early weird fiction probes the very boundaries of reality - the laws and limits of time, space and matter. Here, unimaginable terrors lurk in hitherto unknown mirror dimensions, calamities in ultra-space threaten to wipe clean all evidence of our universe and experiments in non-Euclidean geometry lead to sickening consequences.

In twelve speculative tales of our universe's mathematics and physics gone awry, this new anthology presents an abundance of curiosities - and terrors - with stories from Jorge Luis Borges, Miriam Allen deFord, Frank Belknap Long and Algernon Blackwood

Table of all the important parameters of an icosidodecahedron (having 20 congruent equilateral triangular & 12 congruent regular pentagonal faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume calculated by using HCR's formula for regular polyhedrons. It can be used in analysis, designing & modelling of polyhedrons.

People these days know the Universe as a Whole, because not knowing the edge between This and That. Its secret is the secret of a forgotten body, the seventh in the series of multifaceted as Life, Seven. We know six of it now. But the Ancestors knew all seven. // Люди ныне не знают Вселенной как Целого, не зная грани меж Этим и Тем. Тайна  ее есть тайна забытого тела, седьмого в строю многогранных как Жизнь, Семь. Мы знаем шесть ныне. А Пращуры знали все семь.

In wind load calculations based on pressure measurements, the concept of 'tributary area' is usually used. The literature has less guidance for a systematic computational methodology for calculating tributary areas, in general, and for scattered pressure taps, in particular. To the best of the author's knowledge, there is no generic mathematical equation that helps calculate the tributary areas for irregular pressure taps. Traditionally, the drawing of tributary boundaries for scattered and intensively distributed taps may not be feasible (a time and resource consuming task). To alleviate this problem, this paper presents a proposed numerical approach for tributary area calculations on rectangular surfaces. The approach makes use of the available coordinates of the pressure taps and the dimensions of the surface. The proposed technique is illustrated by two application examples: first, quasi-regularly distributed pressure taps, and second, taps that have scattered distribution on a rectangular surface. The accuracy and the efficacy of the approach are assessed, and a comparison with a traditional method is presented.

All the articles have been derived by the author Mr H.C. Rajpoot by using HCR's Inverse cosine formula & HCR's Theory of Polygon. These formula are very practical & simple to apply in case of any tetrahedron to calculate the internal (dihedral) angles between the consecutive lateral faces meeting at any of four vertices & the solid angle subtended by it (tetrahedron) at the vertex when the angles between the consecutive edges meeting at the same vertex are known. These are the generalized formula which can also be applied in case of three faces meeting at the vertex of various regular & uniform polyhedrons to calculate the solid angle subtended by polyhedron at its vertex.

In this paper, we propose and study two iteration schemes (modified Halpern's type and HS-iteration schemes). Furthermore, it is proved that if two finite families of quasi-nonexpansive mappings satisfy jointly demiclosedness principle, then under appropriate conditions on the iteration parameters, the schemes so introduced strongly converge to the common fixed points of the mappings. Our main results improve and generalize the results in the literature and many other existing results currently in literature.

All the important parameters i.e. inner radius, outer radius, mean radius, surface area & volume of all five platonic solids i.e. regular tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron have been calculated by using HCR's generalized formula for all regular polyhedrons having congruent faces each as a regular n-polygon.

All the important parameters of a small rhombicuboctahedron (an Archimedean solid having 8 congruent equilateral triangular & 18 congruent square faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedrons. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can also be used in analysis, designing & modelling of truncated polyhedrons.

The Golden Ratio(aka “golden mean”, “golden section” or “divine section”: standardly denoted with the capital Greek letter Phi) is the only positive (irrational) quadratic solution of the (minimal polynomial) x^2-x-1=0 equation, so that: Phi=[sqrt(5)+1]/2.
The purpose of this paper is to demonstrate some aspects recently discovered by the author of this paper:
(0) a recursive geometrical method for constructing  Phi (inspired from the Taoist religious iconography), which is more simple and elegant than Odom’s method;
(1) an occurrence of Phi  in the water molecule geometry;
(2) a quasi-Pythagorean property of the (Phi,e,PI)  triad;
(3) some other interesting quasi-exact relations between the elements of the (Phi,e,PI) triad;
(4) the tendency of the human systolic/diastolic blood pressure ratio to stabilize to a value close to  Phi with age

Keywords: Golden Ratio/section/mean (Phi), Odom’s method, Taoist religious iconography, water molecule geometry, quasi-Pythagorean property, human blood pressure

#DONATIONS. Anyone can donate for dr. Dragoi’s independent research and original music at: https://www.paypal.com/donate/?hosted_button_id=AQYGGDVDR7KH2

All the articles, related to three externally touching circles, have been derived by using simple geometry & trigonometry to calculate inscribed & circumscribed radii. All the articles (formula) are very practical & simple to apply in case studies & practical applications of three externally touching circles in 2-D Geometry. Although these results are also valid in case of three spheres touching one another externally in 3-D geometry. These formula are also used for calculating any of three radii if rest two are known & the dimensions of the rectangle enclosing thee externally touching circles.  Here is also the derivation of a general formula to compute the length of common chord, angle of intersection & area of intersection of two intersecting circles.

Learn how to use ArcGIS Desktop to create, edit, manage, display, analyze, and share geographic data
    Use common geo-processing tools to select and extract features
    A guide with example-based recipes to help you get a better and clearer understanding of ArcGIS Desktop

All the articles have been derived by the author by using simple geometry, trigonometry & calculus. All the formula are the most generalized expressions which can be used for computing the volume & surface area of minor & major parts usually each with hyperbolic section obtained by cutting a right circular cone with a plane parallel to its symmetrical (longitudinal) axis.

The numerical simulation using Finite Element (FE) method is a main activity in the design - validation process of mechanical products. Using this method and CAD tools can give an efficient advantage in the CAD - Analysis loop. The geometric model (mesh) used in the FE method is mainly obtained from the CAD model (CSG, B-Rep). In order to reduce the analysis time without bringing precision on computation results, we propose to integrate a new link in the design and analysis interoperability process. This link is named Idealization or simplification of CAD model. So, the automation of the idealization task is important in the CAD-Analysis integration process. In this paper, we propose an idealization approach which mainly depends on the geometry, the solicitations and the behavior of material. According to the analysis goal, the designer can have several alternatives of analysis models which depend on the level of eliminated details and the abstraction level (1D, 2D or 3D). The non manifold models should not be a barrier to the considered analysis model. We start with a positioning of several geometry simplification methods for CAD models in order to generate analysis models adapted with the considered analysis point of view. Based on this state of the art, we present the idealization approach in an integrated design context. Then, we propose our general method of the CAD geometry idealization. This one is based on an original algorithm witch integrate tow alternatives approach (selective approach and automatic approach). In order to emphasize our contribution, we present some examples of idealized form features (parallelepiped, cylinder, and wedge) implemented with OpenCacade code. Finally, an example of mechanical part using implemented form features is developed. The output of the algorithm is some proposed analysis models based on the CAD model of the part.

http://www.designsociety.org/publication/25383/idealization_of_cad_geometry_using_design_and_analysis_integration_features_models

A metric space is a set of points for which we have a notion of distance which just measures the how far apart two points are. The most important and natural way to apply this notion of distance is to say what we mean by continuous motion and therefore to serve as as the foundation of topology (the study of continuous maps and the properties they preserve.

Uno dei possibili approcci al tema della prospettiva parietale è quello di partire con il porsi la domanda: perché nasce la quadratura? Chi decide per questo tipo di decorazione parietale? Come avviene la scelta del pittore di quadro? E, infine, come si muove l’artista dal momento in cui il lavoro gli viene proposto fino a quando la sua opera viene “collaudata”?
Si tratta di indagare le motivazioni del committente e quelle del progetto decorativo, di entrare nel meccanismo che porta dall’esigenza (parete da decorare e volontà del committente) all’opera realizzata (progetto dell’illusione e sua realizzazione in situ). Queste dicotomie committente-artista, ambiente reale-ambiente illusorio, tecnica prospettica-risultato artistico rappresentano la caratteristica prima delle opere che rientrano nel grande capitolo della decorazione quadraturista e implicano la difficoltà, spesso riconosciuta dagli studiosi, di scindere progetto iconografico e progetto prospettico. Se però l’indagine della componente figurativa è stata ampiamente discussa nell’ambito storico-artistico cui naturalmente compete, dove inizia la pertinenza degli studiosi di Storia della Rappresentazione e di costruzione prospettica?
Il testo ipotizza che si possa a ragione affermare che è proprio nel momento in cui la commessa è partita e il committente ha temporaneamente alienato le sue necessità affidandole all’artista che inizia quel capitolo che riguarda da vicino l’approccio degli architetti. È in questo momento, infatti, che entra in ballo la questione progettuale: l’artista comincia a lavorare al “progetto di illusione” che coinvolge l’interpretazione stessa dello spazio architettonico in cui l’opera sarà realizzata, che implica una sua trasformazione, sia pure con i soli mezzi dell’illusione, una astuta interpretazione della fruizione dello spazio reale che prelude al coinvolgimento guidato dell’osservatore e delle sue percezioni attenuto atraverso lo strumento prospettico utilizzato nel pieno della sua espressione.

KEY WORDS: Quadratura, Quadratrurismo, Scienza della Rappresentazione, prospettiva, restituzione, superfici, Vallombrosa, Abbazia, raddrizzamento

Gelişen imalat teknolojilerinin yeni bir kolu olan eklemeli imalat kendi içinde birçok farklı üretim alternatifine sahiptir. Çalışmada eklemeli imalat yöntemlerin üretim teknikleri, malzeme alternatifleri ve daha iyi sonuçlar alabilmek adına hesaplamalı analizler yapılmaktadır. Geniş malzeme yelpazesine sahip eklemeli imalat teknolojisi ürün kalitesine adına yapılabilen birçok araştırmaya çalışma içerisinde sonuçları ile mevcuttur. Yüzey ve yapı karakterlerini iyileştirmek adına yapılabilecek farklı üretim kombinasyonları çalışma içinde detaylı bir şekilde incelenmektedir. Muazzam geometri toleransıyla yeni dünyada çok fazla tercih edilmeye başlanan eklemeli imalat teknolojileri büyük bir ivmeyle geliştirilmektedir. Özellikle metal tozları ile üretilen ürünlerin makine teknolojilerinde büyük talep içerisinde bulunmaktadır. Ürün kalitesini artırmak adına topoloji, yapı statiği ve termal analizler büyük imalat yüklerini daha optimize etmek için yapılmaktadır. Çalışmada tasarlanan fan kanatların SLM metoduyla üretilmesi adına tasarımı ve üretim maliyetlerini düşürmek için yapı ve topoloji analizleri yapılmıştır.  .

English!

Additive manufacturing, which is a new branch of developing manufacturing technologies, has many different production alternatives in itself.  In the study, production techniques of additive manufacturing methods, material alternatives and computational analyzes are carried out in order to get better results.  Additive manufacturing technology with a wide range of materials is available with the results of many studies that can be made on behalf of product quality.  The different production combinations that can be made to improve the surface and structure characters are examined in detail in the study.  Additive manufacturing technologies, which are very much preferred in the new world with their enormous geometry tolerance, are being developed with great acceleration.  Especially, products produced with metal powders are in great demand in the machine technologies.  Topology, structural statics and thermal analyzes are performed to further optimize large manufacturing loads in order to increase product quality.  In order to produce the designed fan blades with the SLM method, structure and topology analyzes have been made in order to reduce the design and production costs.  .

This document presents an evolved Girih tile set beyond the five Girih Tiles popularized by Lu and Steinhardt. Additional tiles have been presented in multiple papers before, but here, a wider set is officially defined. I refer to this as the evolved tile set or evolved Girih tile set. It includes the adapter tiles, which also have been presented before, but not defined as a group or part of a tile set. Adapter tiles differ from the five Girih Tiles as the latter only have equilateral sides. This paper presents one category of adapters containing tiles having at least one side with the length of Φ. The aim is to establish non-equilateral sides as a necessary concept to enable tiling of a greater number of existing Islamic geometric patterns. This opens up for future introductions of other categories of non-equilateral tiles.

Approximate value of solid angle subtended by an elliptical plane at any point lying on the axis passing through the centre can be calculated by using HCR's Approximation Formula which is applicable on any plane having at least two axes of symmetry & factor of circularity closer to unity for a given location of the observer in the space. It is equally applicable for  rectangle, square, rhombus, ellipse, circle & all other symmetrical planes having at least two axes of symmetry.

(english, then french)Psychical reality in the R diagram (annoted diagram appearing in Lacan's Writings ("Écrits") titled: “Of a preliminary question to any possible treatment of psychosis”). What is being developped is expanded on my comments of the R diagram in my psychiatric thesis addressing the issue of psychotherapy of psychoses. It is advisable to download and open it in another window for a better monitoring of my comment, the non-developed R Diagram appears in the original text, as well as the developed R Diagram drafted by myself based on the notes appearing in the text of Lacan.
............................................................................................................................................... La Réalité psychique dans le schéma R (schéma commenté figurant dans le texte des Écrits de Lacan intitulé : « D'une question préliminaire à tout traitement possible de la psychose »)
Ce qui suit est développé à partir de mon commentaire du schéma R dans mon Mémoire de Psychiatrie sur la psychothérapie des psychoses. Il est recommandé de télécharger et d'ouvrir dans deux autres fenêtres, pour mieux en suivre le commentaire, le Schéma R original de Lacan, et le Schéma R développé par moi d'après le indications de Lacan. Ces deux schémas sont dans les deux annexes qui suivent immédiatement cet article.

This paper objectively presents a provable construction of generating a length of magnitude  , as the geometrical solution for the ancient problem of doubling the volume of a cube. Cube duplication is believed to be impossible under the stated restrictions of Euclidean geometry, because the Delian constant  is classified as an irrational number, that was stated to be geometrically irreducible (Pierre Laurent Wantzel, 1837) [1]. Contrary to the impossibility consideration, the solution for this ancient problem is theorem  , in which an elegant approach is presented, as a  refute to the cube duplication impossibility statement. Geogebra software as one of the interactive geometry software is used to illustrate the accuracy of the obtained results, at higher accuracies which cannot be perceived using the idealized platonic straightedge and compass construction.

El trabajo hecho en esta tesis fue estudiar algunas propiedades de la topología simpléctica en particular el comportamiento local de variedades simplecticas, para eso se
entendió la demostración del teorema de Darboux, el enunciado de este es interesante
por que afírma que dos variedades diferenciables con estructura simplécticas localmente
no tienen invariantes topológicas, eso quiere decir que al momento de analizar
dos variedades simplecticas y querer compararlas, no serviría un estudio local por
que el teorema nos afrimaría que son idénticas y que se comportan de la misma manera
alrededor de cualquier punto, esto nos indica que tenemos que hacer un estudio
global para diferenciar algunas propiedades topológicas y geometricas. Si el estudio
que se quiere hacer de cualquier variedad simpléctica es puramente local basta con
analizar la variedad simpléctica (R2n; !0), por que el teorema de Darboux nos afírma
que son simplectomorfas localmente. Esto simplifíca enormemente el trabajo, dado
que se conocen muchas cosas de esta variedad diferenciable.
Lo interesante de este trabajo es la forma en como se estudio la demostración del
teorema de Darboux, basándose en el trabajo de Moser sobres campos vectoriales
y formas diferenciales, esto simplíca y pone con mas claridad la demostración, de
tal manera que el teorema de Darboux queda como aplicación inmediato de esté
trabajo. Lo que no se observa en otras demostraciones hechas para esta teorema.

La presente tesis doctoral, es un esfuerzo por aportar una propuesta de hacer diseño arquitectónico ante las exigencias actuales de la complejidad. La discusión ha sido extensa en cuanto a que el diseño debe ser complejo, el problema es, que desde nuestro campo de conocimiento, no se ha dicho satisfactoriamente cómo. El objetivo inicial esperado en la investigación, consistía en la vinculación compositiva espacial entre la música y la arquitectura, trascendiendo su dimensión simbólica y sus entidades métricas; sin embargo, ante la falta de una estructura teórica adecuada al contexto del diseño, fue necesario iniciar la construcción de un argumento categórico generalizante, que integrara aspectos conceptuales abstractos, y aspectos materiales de realización física, en un lugar común (τoπoς).
Este argumento se convierte formalmente, en un modelo semántico basado en los sistemas evolutivos de memoria (Ehresmann, Vanbremeersch, 2007), que genera un patrón complejo a nivel conceptual, y que funciona como una metaestructura, en conjunción con la generación de ontologías computacionales y estructuras gráficas jerárquicas, como significantes para la representación formal del conocimiento y son elementos conformadores para la resolución de los problemas representativos de la complejidad en el contexto del diseño. Esta base de conocimiento integra herramientas
y métodos de inteligencia artificial, mediante el uso de Sistemas Clasificadores de Aprendizaje (L.C.S.)(Drugowitsch, 2008), (Urbanowicz, 2009), que basadas en el universo de  conocimiento dado, posibilitan razonamiento.

My presentation in Kanazawa aims to bring a renewed interest and a new nexus of activities around William Huff's work, especially as an exercise in transformational thinking in the field of architectural education. After his retirement, professor Huff donated a beautiful collection of his students' work to the HfG-Archiv in Ulm, Germany, which inspired us at the Institute of Art and Design in Vienna to conduct a semester assignment for 450 students around the topic of Parquet Deformations (P.D.). The open search for strategies to transfer the spatiotemporal idea of a planar P.D. into 3D led to fascinating results. Hereby we discovered a field of formal research that opened the students' eyes for two and three-dimensional relationships and made them enthusiastic and sensitive for spatial transformations. This is a brief report on how we conducted the exercise, as a fruitful exercise for the basic design education of young architects and designers. William Huff, experimenting with his students on parquets consisting of one geometrical shape only, discovered a way to transform, slowly and subtly, one geometrical shape into another and set up strict rules for the assignment: the parquet had to be made up with one geometrical shape only (no overlaps, no gaps). The allowed lattices were square, rhombic (the rhombic has to be a 60°-120° rhombus) or regularly hexagonal. Parquet Deformations (P.D.s), designs, produced for over three decades in a curriculum of basic design studies, are rooted in two analytical disciplines: monohedral tiling from geometry; continuous deformation from biological morphology (after D'Arcy Thompson). The HfG-archive also hosts intricate 3D-models made under Huff's guidance. Especially the "trisections of the cube" and William Huff's comment