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In the seventeenth century, Guarino Guarini, mathematician and architect, affirmed that architecture, a discipline that primarily deals with measures, relies on geometry: therefore, the architect needs to know at least its basic principles. On behalf of Guarini’s words, we designed a set of interdisciplinary teaching experiences, between mathematics (via a calculus course) and drawing (via our Architectural Drawing and Survey Laboratory courses) that we proposed to first-year under graduate students studying for an Architecture degree. The tasks concern mathematical and representational issues about vaulted roofing systems and are based on the use of physical models in conjunction with digital tools, in order to make the cognitive geometric process more effective, thus following a consolidated tradition of both disciplines.

L’attività descritta in questo articolo si propone di stimolare, attraverso la pratica della piegatura origami, una riflessione metacognitiva sul tema dell’apprendimento e consiste in un’azione creativa che consente di sperimentare alcune delle difficoltà comuni nel percorso di studi universitario, individuando fin da subito strategie utili per affrontarle.L’esperienza, da svolgersi nelle prime settimane di lezione, mette in luce l’importanza di sviluppare un atteggiamento consapevole nei confronti dello studio e affronta due temi cruciali per l’efficacia dell’apprendimento: il ruolo della curiosità come spinta motivazionale a imparare cose nuove e l’importanza della fiducia nel docente e nelle sue proposte didattiche. In questa sede si presentano i risultati di un’applicazione nell’ambito di un corso universitario di matematica, benché l’attività, che per sua natura insiste più sul piano metodologico che non su quello strettamente disciplinare, possa essere trasferita efficacemente anche ad altri insegnamenti.

To make the geometrical cognitive process more interactive, we produced teaching aids (tangible models, graphic tablets) that help students in visualizing their geometrical-analytical investigations of the architectural artifacts and enhance their spatial prefiguration and critical form-reading skills, three-dimensional thinking and geometrical reading of shapes. Then, we looked for a medium suitable to create simple three-dimensional models, not only observable, like virtual models, not only tangible, like physical models proposed in the design studios, but also dynamic, using multiple media and languages in the same training message. As an example, we present here an interdisciplinary lesson between Calculus and Architectural Drawing and Survey Laboratory about developable surfaces, experimented on first year students of the bachelor program in Architecture. The lesson is based on the use of a graphic tablet and some origami inspired models: it summarizes the geometric description of a pyramid and a cloister vault of equal height and equal orthographic projection on the horizontal plane. We saw that tackling the same topic in both teaching contexts is not a useless overlap, but a stimulus to compare different languages and methods. 2D and 3D paper models of artifacts—and of projective reduction from 3D to the plane—aid spatial intuition and the subtle exercise of controlling mental images which replace artifacts, turning 3D configurations into signifying images. Moreover, this experience stimulates reading and evaluation of the drawn geometry (ruled surfaces, projections, developments), increasing critical sense in reading the built environment.

It is well known that a way to interest children and to teach new topics is playing with them. We can involve children, but not only, using educational games available on the market or construct with them all the pieces and the rules they need to use to develop new games. In this paper we propose an educational project involving maths and representation The principal aim is to explain the methodology we use in the construction of original evolutions of the traditional windmill origami base in order to obtain connectable pieces for tessellation (3-4-5-6- edges tesserae equipped with winglets and pockets). Then we propose how to use the produced tesserae, during both the folding or connecting process, in order to: • learn geometric language through the modeling of basic paper module; • educate to the shape reading by decomposing and recomposing it; •educate to understand the relationship between objects, symbolic representation and descriptive representation The chosen media is origami, because it is able to surprise participants -children for magic and older students for effectiveness-: it is the ideal tool to show geometry in a seemingly informal but also extremely rigorous way. Moreover, the use of the colored paper allows students to fold, in a cheap way, a lot of funny pieces which can be dynamically and creatively aggregate. The math lesson starts from the folding passages and continue during the explorations of some geometric topics in the puzzle combinations: geometric properties of polygons, as symmetries, equiestension and isoperimetric problems, can be investigate getting out relationship among tessellation, drawing and maths. For oldest students, teacher can discuss also the construction methodology. The activity offers the possibility of making concrete reasoning on the relationships between geometric figures: the modular elements, made during practice, become cognitive artifacts to prepare the mind to welcome intuitions and discoveries. The use, by the teaching staff, of a specific language calibrate according to the participants, to support direct observation, leads to the development of geometrical abilities that become the basis for subsequent implementations throughout the entire educational path, thus becoming part of a vertical distribution of knowledge. The description of complex shapes during a tangram-based game brings students to read shapes from an outline and viceversa, allows them to create new puzzles, with different levels of complexity and works on drawing as a graphic trace. The possibility of expressing the experience by a double point of view, math & Art, with the same language let bring us in an exploratory journey in concrete Geometry. Until now, we proposed these activities in primary and middle school and in training classes for teachers who tested the activity in their schools finding a good interest from participants. We will test this game also in occasion of “Il Salone del Libro di Torino”, for students and generic public as the game can be adapted to adults too, offering games with incremental difficulty.

Our main purposes are the disclosure of certain basic topics of geometry to a broad audience with the aim of enabling people to read geometric shapes that surround us, both from a mathematical and architectural point of view. In the events that we organize, we always leave ample margin to material testing, following the learn-by-doing philosophy. In particular we use origami as a medium to convey some knowledge about shapes. We present various examples addressed to different audiences.

In this paper, I describe some activities proposed to students of the General Mathematics course at three years Architecture Degree of Politecnico di Torino. Students attending the course show some difficulties in studying the topics, mainly due to the intrinsic difficulty of topics and to the lack of interest have they in them. Here are some examples of activities proposed in this course and enthusiastically received from the students. For this purpose, different tools are used: from the ancient origami art to modern software as GeoGebra. This approach often surprises students and shows them that they can see and use mathematics in many different ways.

In this paper we describe how, starting from our experience of Geometry's teaching in two different academic contexts (mathematic and drawing), we identified together common didactics tools in order to educate the reading of the geometric shapes and their properties with particular attention to the architectural cases. To do this we use 3D models (for example origami models) and gaming activities, both with students and with trainers and our lessons are held not only in the classroom but in public places such squares or cultural sites. Today, we use our teaching idea to train the trainers to understand how to teach geometry in each kind of school level in a new way, using a tangible geometry.

In this paper we investigate degree two curves of arbitrary codimension. This requires to study also ropes supported on a line. After establishing several characterizations of ropes supported on a line we describe their homogeneous ideals and Hartshorne–Rao modules; in particular we characterize the arithmetically Buchsbaum ropes. Then we describe the even Gorenstein liaison classes of the non-arithmetically Buchsbaum, non-degenerate ropes generalizing a well-know result of Migliore on double lines of codimension two. The result relies on the explicit description of certain arithmetically Gorenstein curves with maximal tangent spaces at each point. As a consequence, we can decide if two curves of degree two belong to the same Gorenstein liaison class. Finally, we show as an application how ropes can be used in order to construct quasi-extremal curves.

Given a smooth k-variety Y (where k is a field of arbitrary characteristic) and a linear systemL on Y we study the dimension of the singular locus of the general element ofL, both inside and outside the base locus B ofL. We interpret these results from the point of view of the transversality theory, and we improve a result by Speiser about the not too ramified morphisms. Moreover, we show that our results can be applied in some cases where a criterion by Zhang, for the smoothness of the general element ofL, fails.

In this paper, we study the curves C in P n , of degree d and genus g , with extremal Rao function in positive degrees, and non degenerate general hyperplane section. We describe their total ideal and various properties of the generators of the ideal. Moreover, we characterize these curves as intersection of two aCM curves of maximal genus whose union in aCM of maximal genus, and we completely describe the Rao function of these curves in negative degrees, too.

Let k be a field of any characteristic and not necessarily algebraically closed. In this paper we extend some results of the transversality theory to larger families of k-schemes and of k-morphisms and to all the local geometric properties satisfying some given axioms (for example: geom. regular, geom. reduced, geom. normal and the properties Sr and Rs). As an application, we obtain a generalization of the well known Bertini type theorems for local geometric axiomatic properties.

In this paper, we study double structures supported on rational normal curves. After recalling the general construction of double structures supported on a smooth curve described in [11], we specialize it to double structures on rational normal curves. To every double structure we associate a triple of integers (2r, g, n) where r is the degree of the support, n ≥ r is the dimension of the projective space containing the double curve, and g is the arithmetic genus of the double curve. We compute also some numerical invariants of the constructed curves, and we show that the family of double structures with a given triple (2r, g, n) is irreducible. Furthermore, we prove that the general double curve in the families associated to (2r, r + 1, r) and (2r, 1, 2r−1) is arithmetically Gorenstein. Finally, we prove that the closure of the locus containing double conics of genus g ≤ −2 form an irreducible component of the corresponding Hilbert scheme, and that the general double conic is a smooth point of that component. Moreover, we prove that the general double conic in ℙ3 of arbitrary genus is a smooth point of the corresponding Hilbert scheme.

We study families of ropes of any codimension that are supported on lines. In particular, this includes all non-reduced curves of degree two. We construct suitable smooth parameter spaces and conclude that all ropes of fixed degree and genus lie in the same component of the corresponding Hilbert scheme. We show that this component is generically smooth if the genus is small enough unless the characteristic of the ground field is two and the curves under consideration have degree two. In this case the component is non-reduced.

In this paper, we study the Hilbert scheme of non degenerate locally Cohen- Macaulay projective curves with general hyperplane section spanning a linear space of dimension 2 and minimal Hilbert function. The main result is that those curves are almost always the general element of a generically smooth component Hn,d,g of the corresponding Hilbert scheme. Moreover, we show that the curves with maximal cohomology almost always correspond to smooth points of Hn,d,g.

In this paper, we present a method to inductively construct Gorenstein ideals of any codimension c. We start from a Gorenstein ideal I of codimension c contained in a complete intersection ideal J of the same codimension, and we prove that under suitable hypotheses there exists a new Gorenstein ideal contained in the residual ideal I: J. We compare some numerical data of the starting and the resulting Gorenstein ideals of the construction. We compare also the Buchsbaum-Eisenbud matrices of the two ideals, in the codimension three case. Furthermore, we show that this construction is independent from the other known geometrical constructions of Gorenstein ideals, providing examples.

Let L be a line contained in a Grassmannian variety G. A d-rope C _ G supported on L is a locally Cohen-Macaulay curve of degree d with Cred = L and (IL,G)2 _ IC,G. We characterize the d-ropes C supported on L and embedded in G. In some cases we describe also the vector bundles on such a rope C. Finally, we describe the parameter spaces for ropes embedded in G.

Here we show the existence of strong restrictions for the Hilbert function of zerodimensional curvilinear subschemes of Pn with one point as support and with high regularity index.

We show that the set of the homogeneous saturated ideals with given initial ideal in a fixed term-ordering is locally closed in the Hilbert scheme, and that it is affine if the initial ideal is saturated. Then, Hilbert schemes can be stratified using these subschemes. We investigate the behaviour of this stratification with respect to some properties of the closed points. As application, we describe the singular locus of the component of Hilb4 z +1 ℙ4 containing the ACM curves of degree 4.