Architectural Geometry

In any triangle, the perpendicular side bisectors meet the corresponding internal angle bisectors on the circumcircle. If we take those three points as the vertices of a new triangle and repeat the operation indefinitly, we end up in the limit with a par of equilateral triangles whose sides are parallel to the sides of the Morley triangle of the initial triangle.

Introduction
The cost of building dams is very high and their failure can be hazardous. On the other hand, they are vital for every country as freshwater storage. Deterministic and traditional algorithms can not answer the multidimensional and complex problems of dam construction, and it is necessary to use hybrid methods based on probabilities. The problems of fluid movement in their nature have a complexity that modeling and finding requires using an advanced algorithm that can finally interpret its non-deterministic nature. Earth dams have a porous, multiphase, and complex medium, and the hydraulic and mechanical variables in different parts are associated with uncertainty. For this reason, in recent years, the regulations for the design of dams have been reforming in the direction of applying non-deterministic and probabilistic variables in the calculations. A probabilistic engineering view leads to a more realistic understanding of design than deterministic approaches. In the research, artificial intelligence (AI) methods have been used to analyze the data, which provides a predictive model of behavior for seepage discharge flow through the earth dam. In general, the present research has two purposes: a) to estimate the effect of uncertainty of the hydraulic conductivity dam on seepage discharge and b) to provide a model to estimate seepage discharge in a dimensionless way with the gene expression programming (GEP) and support vector machine (SVM) methods.

Materials and Methods
Monte Carlo simulation (MCS) with 2000 iterations was executed for stochastic analysis. The first step of the Monte Carlo simulation is the choice of the deterministic performance function. In the second step, the input variables were defined to the performance function and the probability distribution for variable/variables. By repeating the process n times, n random answers were extracted for the resulting problem, and finally, probability density function (PDF) and cumulative density function (CDF) graphs were drawn for the results. In the Fortran code of this research and to check the convergence, the hydraulic heads were compared to achieve the difference obtained in iteration n with the obtained value in iteration n-1, and if the difference is less than the tolerance error, then the program stops. In the next section of the algorithm, the obtained data (from the repeated execution of the MCS) are converted into a model for a description relationship between the effective seepage discharge (ESD) and the input variables by using the metaheuristic methods that include; gene expression programming (GEP) and support vector machine (SVM). After GEP and support vector regression (SVR) modeling the predicted and observed results were compared by statistical indexes such as MSE, RMSE, MAE, and Correlation coefficients.

Results and Discussion
The different models of earth dams were implemented in the Fortran program, and the average and standard deviation of the seepage discharge flow in the uncertainty state were obtained. To determine the relationship between the ESD value, indicators had been defined that these parameters used for the Gene Expression Programming model include; Kx/Ky, W/B, Bd/B, Bu/B, Hdam/B, Hu/Hdam, and Hd/Hu. These were the factors influencing the seepage discharge of the earth dam, and the discharge component is also defined as the effective seepage discharge (ESD) in a dimensionless manner. Kx and Ky are soil permeability in the direction of the horizontal and vertical axes respectively (m/s), W is the width of the crest, B is the width of the base of the earth dam, Bd is the horizontal distance of the dam tip from the downstream side from the crest, Bu is the horizontal distance of the dam tip from the upstream side crest, Hdam height of the dam, Hu height of the reservoir level, Hd water height downstream of the dam, all the variables are in meters. By increasing the Kx/Ky ratio of horizontal to vertical hydraulic conductivity by 49%, the Effective Seepage Discharge increases by 14%. If the horizontal variable of permeability is increased by 25%, the ESD rate increases by 4.56%, similarly, if the vertical variable is increased by 25%, the ESD decreases by 4.72%.

Conclusion
After finite element analysis, and modeling with two methods of gene expression programming (GEP) and support vector regression (SVR), the statistical analysis of the methods showed that the two calculation models had a good prediction of the ESD with a correlation coefficient above 0.9. Vertical hydraulic conductivity (Ky) has a greater effect on the ESD rate than horizontal hydraulic conductivity (Kx). The results of the geometric investigation of the dam also show that the increase in the ratio Hdam/B has a direct impact on the ESD and also the lower the slope downstream of the dam leads to the lower the ESD. The statistical analysis was used to compare the results of the data obtained from Fortran output for SVR and GEP models. In general, the SVR model is closer to the model resulting from the Fortran code rather than the GEP model, and it has a low root mean square error (RMSE) and a high correlation coefficient.

Si farà, in questo breve lavoro, riferimento alla cosiddetta Teoria delle Geometrie Formali, cui si rimanda per un eventuale approfondimento (comunque non indispensabile) per seguire il processo di costruzione geometrica della fabbrica di San Bernardino in Chieri 1 . Non posso comunque esimermi dal richiamare alcuni concetti chiave inerenti lo stretto rapporto che lega la geometria al processo di ideazione dell'essenza e realizzazione della sostanza di un'architettura, rapporto spezzato raramente e in quei soli casi ove il "gesto" ha potuto esprimersi liberamente: un caso su tutti, la cappella inventata da Le Corbusier a Ronchamp 2 . In primis, la geometria è, dai tempi di Euclide in poi, l'espressione di una struttura logica che, attraverso i suoi assiomi e postulati, i suoi teoremi e dimostrazioni, descrive in maniera utile una possibile realtà: il metodo deduttivo che le è proprio garantisce il controllo costante degli effetti derivati da cause date. In secondo luogo, la strutturazione geometrica dello spazio, così come delle forme che lo generano, consente facilità di orientamento all'interno dello stesso: sembra, questa, una prerogativa propria alla razza umana, visto che, pur concepita con fini diversificati, la geometria è presente in ogni evo e in ogni cultura che palesi un sia pur contenuto grado di civiltà. Il richiamo costante di allineamenti, squadri e livelli, per dirla con Vitruvio 3 , rende fluida la percezione della forma: il perché è più questione psicologica e fisiologica

Este artículo explora la figura del heptagrama regular desde una perspectiva multidisciplinaria que abarca la matemática, la historia, el arte y la espiritualidad. Se analiza su geometría, destacando su relación con el heptágono regular y las dificultades de su construcción clásica con regla y compás.
A través de un recorrido histórico y simbólico, se revela la presencia del número siete en diversas culturas y tradiciones, desde la antigüedad babilónica hasta el ocultismo moderno. El texto profundiza en las propiedades métricas del heptagrama, sus ángulos, proporciones y su papel en teselaciones no euclidianas.
Como aporte práctico, se incluyen dibujos nemotécnicos que permiten realizar los dos tipos de heptagramas en un solo trazo sin levantar la mano, a modo de juego de coordinación mental y motriz.
Se propone al heptagrama como un puente entre ciencia y espiritualidad, una figura que inspira tanto a matemáticos como a artistas por su riqueza simbólica y estética.

Free-form timber structures are currently a high interest topic that represents a limited niche within the AEC industry and a resource intensive choice for building. This research explores an alternative to current technologies for their production through “Curve-Fit”, a building system that uses flat-packed timber modules that can be transformed through curved-folding. Transformed modules are connected along curved creases to form load bearing structures. A set of strategies were developed for the geometric design, structural validation and detailing of the system. These were wrapped in an integrated feedback workflow for the informed coordination of multiple design objectives. The system was tested with a full scale demonstrator, proving that load bearing structures can be produced with the Curve-Fit system. The proposed strategies have the potential to increase production efficiencies and reduce resource consumption in the production of free-form timber structures.

En esta ocasión presentamos la construcción del carro de guerra completo mediante un dibujo en 3D, coloreando las placas planas, de madera en su dibujo, demostrando así que Leonardo intuyó y dibujó una superficie reglada entre una elipse y una circunferencia.

Chapter contribution to book accompanying the exhibition "Wonders of Creation: Art, Science, and Innovation in the Islamic World," organized by Ladan Akbarnia for the San Diego Museum of Art, inspired by Qazwini's Ajaib al-Makhluqat, written in the 13th century..

This paper examines the geometry of the King’s Chamber, the granite coffer within it, and the wider pyramid field at Giza, with the aim of identifying underlying patterns of numerical and harmonic design. The elevation of the King’s Chamber floor corresponds precisely to values derived from π, while the chamber’s internal diagonals and proportions resolve into integer-based harmonics. The coffer, far from being a mere container, reveals volumetric and dimensional relationships tied to canonical numbers, the lunar diameter, and the Great Pyramid’s base length through the expression 22/7. Extending beyond the pyramid’s interior, large-scale measurements across Khufu’s and Khafre’s monuments reproduce Earth’s equatorial circumference and constants such as √3 when set in relationship.

These findings suggest that the builders of Giza deliberately embedded terrestrial and celestial references into their architecture. The result is a synthesis of structure and symbolism: monuments that operate simultaneously as tombs, temples, and repositories of mathematical knowledge.

Arches, vaults, and domes are among the most fundamental forms used since antiquity. Gothic and classical Ottoman architecture were milestones
in structural elegance and material efficiency. In the modern period and afterward, a design approach based on chain curve theory created the most
efficient forms for curvilinear structures, such as shells and bridges. As this method is digitized, historical forms are being customized to novel
materials. However, in many current buildings, form, and material are not harmonized, which contradicts the principles of efficiency and
sustainability. In this study, the problem is addressed to the increasing theoretical focus of building technology courses in architecture. At the same
time, practices are limited to calculation and analysis, and basic approaches such as intuition and practice are neglected. In the study, three
different arch models have been designed, executed, loaded, and tested according to chain curve theory and construction principles. This process
has been managed collaboratively by the professor and students, aiming to achieve a learning and teaching experience in line with the philosophy
of lifelong learning, learning to learn, and co-education.
Firstly, the arch designs with chain curve have been elaborated. Then, the structural effectiveness of some arch types common in Türkiye have been
compared according to their ability to conceal the chain curve within the minimum cross-section. Afterward, the same span of three round arches
with different forms, sizes, and construction characteristics have been designed. These three arches have been executed in 1/3 scale brick with
different brickwork and mortar qualities. After finishing the execution, molds have been removed by burning or wetting. Finally, the models have
been loaded with clay bags (static) and stepping (dynamic) on them. Collapse morphology of the arches have been observed, and their compatibility
with the chain curve has been evaluated. Consequently, the first arch collapsed because of the sliding of the bottom and lack of tie-rod and buttress.
In contrast, the second arch collapsed because of the detachment of the mortars due to the lower-positioned tie-rod and the cascading of the
brickwork at the bottom.
The following findings have been achieved from the executions:
• Regarding form, a cascading arch and its keystone raised following the chain curve, with an adequate third dimension, is appropriate
and efficient.
• Regarding style, among the classical arches, five-centered and pointed arches are the most effective, while the shouldered arch is the
weakest.
• Regarding construction, qualified materials and methodical brickwork are required for a tie-rod positioned where the chain curve
approaches the arch line and is connected with a vertical element against shear force.
As a result of intuition, observation, construction, and loading, the student group (Izu), who executed the arches, have been given surveys about the
impact of mold, brickwork, mortar, tie-rod, and chain curve on the quality and structural behavior of the arch models. The survey results have been
compared with scientific findings to test the learning level. The process has been lectured to 174 students at another university (Samsacu) who had
taken the course from the same professor, and the same survey has been conducted. With the data collected, the accuracy of the questionnaire has
been improved, and a comparison and evaluation of learning has been realized by obtaining different opinions from a non-executor group.
According to the average of the answers, it has been certified that 1 out of 3 students in Samsacu and more than half in the Izu gave a correct and
specific explanation. For the main factor that destructed the first arch, 3 out of 5 from the Izu and 1 out of 5 from the Samsacu, respectively, and
approximately; for the second arch, both groups gave similarly correct and specific answers (30% and 25%). The low accuracy rate in the
practicing group has been attributed to the fact that the destruction of this carefully erected arch was sudden and unexpected; therefore, the
behavior could not be observed. It has been concluded that those who did not witness the practice. However, they followed the process
enthusiastically and could not understand what they had learnt because they were away from physical contact.
As a result, the following outcomes have been obtained in the study:
• Interactively creating a model accelerates learning and mobilizes students for learning to learn.
• Facilities of the university and the social potential of student clubs alleviate the time and energy demands of such practices.
• Cooperate with sponsoring companies to supply the necessary equipment to minimize costs and offer students internships and job
opportunities.
• By findings and models, course materials can be enhanced, and future students can benefit from this experience.
It is hoped that this study will contribute to the developing traditional and innovative models with different forms and constructions and
strengthening the practical aspect of architectural education.

Gothic architecture was despised from the Renaissance until the 18 th century. It was considered barbaric and deformed. This appreciation changed in the course of the 18 th century and by the mid-century some educated gentlemen began to use Gothic motifs in their mansions. At the same time some antiquarians, architects and amateurs began to speculate about Gothic architecture. One of the central themes was its origin and it was commonly agreed that this origin was linked to the appearance of the pointed arch.

The main purpose of this paper, which belongs to a research project started in 2018 [1], is that of investigating the geometry and stability of the dome of the Diana’s temple, which survived with its half part along many centuries. Opus caementicium in fact allowed the Romans to build domes and vaults of great amplitude, thanks to an expert use of rubble, mortar, laying techniques and, above all, geometrical parameters, especially as regards to the appropriate outline of the profile and the choice of span and thickness to ensure stability. In this respect De Angelis d’Ossat [2] affirms: “We do not know if the dome’s profile – graphically never detected – corresponds to either one or more curves (catenary, parabola, circle’s arcs), nevertheless the typical curve trend by us defined as ovoid, can be usually observed in the vaults of all the ancient Orient, by starting from the rare significant Egyptian examples of barrel vaults”. After in time, Rakob [3] tried to solve the debated que...

An approach to Descriptive Geometry from digital graphic tools is presented, in which the form of graphic interaction that arises from these tools is analysed, and the importance of their associated automatisms and procedures is discussed. We review the classic idea of graphical thinking to introduce the concept of "augmented graphical thinking": this is linked to the new paradigm of simulation, which is understood as an evolution of the previous paradigm based on representation. Understanding this paradigm shift allows us to approach Descriptive Geometry with digital graphic tools from an in-depth gnoseological foundation, but not without observing certain contradictions that must be analysed and discussed. By focusing on the so-called digital automatisms, we demonstrate that these form part of a long tradition in the history of Graphic Expression, since, like other drawing mechanisms machines designed in the past, they involve the search for the "shortest way" to solve the problem of image construction. Lastly, in the conclusions we address the inevitable connection between Descriptive Geometry and Digital Fabrication.

Back in the 1960s the methods used to record carved stone decoration were, to take rubbings, making plaster casts, latex moulds or paper squeezes. Rubbings are limited to flattish surfaces with incised decoration or script, they do not easily record changes of surface irregularities or relief. Plaster casts have the advantage of providing a durable and precise three-dimensional replica, but they are heavy and depending on the size they can be cumbersome, and unless there is an overriding reason for choosing them their application in the field is restricted. In contrast, latex moulds produce precise light weight three-dimensional replicas that can be rolled up and carried in a pocket, a down side is that quantities of latex are required and sometimes there can be slight discolouration left on the stone surface. Paper squeezes have a number of advantages: the simplicity of the materials required, paper and water; the ease of transporting paper, a bucket and small stippling brush is not onerous; being paper they can be drawn on to confirm what the object was, as in Fig. which is a defaced inscription high up on the south face of the north qanatir or colonnade in the Haram as-Sharif, Jerusalem 1 ; when still slightly damp a squeeze is ready to be removed from the stone and it does required to be laid flat in the sun to dry completely, winter weather is difficult; being composed of two or three layers, they can be separated enabling three attempts to retrieve any lost details; the capturing of minor three dimensional features depends on the skill of the person making the

In hillside townships, the principles of Dynamic Urbanism play a vital role in shaping the built environment and ensuring the well-being of residents. This article explores key aspects of urbanism, focusing on path planning, service provision, housing design, community spaces, sustainable development, safety measures, and the importance of geometrical division in plots and sectors. By carefully considering these factors, hillside townships can be transformed into resilient and livable environments that cater to the needs of their residents. Road Planning and Connectivity: Efficient path planning is crucial in hillside townships to ensure connectivity and accessibility. The unique topography of hilly terrains requires careful consideration when designing paths. Well-designed footpaths, staircases, and ramps can provide safe and convenient routes, promoting walkability and reducing dependence on motorized transportation. The geometrical division of plots and sectors plays a significant role in determining the layout of paths, ensuring efficient and logical connections between different parts of the township. Over the years, paths often grow to form the connecting roads and then they become the spine.

In this article, Shailesh Shirali begins with a seemingly simple question but develops the answer into not one, but four different proofs! While the content focuses on mathematics that has many applications, some of which are mentioned here, the multiplicity of proofs is an added draw, helping the teacher to illustrate innovative ways of thinking and connections across approaches

Regular polygons have been known for a long time to be able to be inscribed in a circle, where all the polygon's vertices lie on the circle's circumference. A circle is an infinite set of points that are all equidistant from a central point, and a regular polygon will share as many points with that circle as it has sides. It is also known that one can construct a regular tangential polygon by having each of its sides be tangent to the circle. While these relationships are well-established, the specific properties and comparisons of their respective radii, particularly when looking at a regular tangential polygon and a regular circumscribed triangle, offer a rich area for study. This paper compares the radii produced by these two distinct geometric configurations, analyzing how the number of sides, n, and the nature of the polygon's relationship to the circle-whether it is tangent to the interior (inscribed) or its sides are tangent to the exterior-affect its radius. Through a generalized formula for regular polygons with n sides, we mathematically demonstrate the relationship between the in-radius and circumradius and show how this relationship varies with n.

We present an alternative geometric derivation of the triangle area formula expressed solely in terms of side lengths a, b, and c. Using elementary geometry and the Pythagorean theorem, we systematically derive the formula as shown in the pdf. This approach provides clear geometric intuition connecting basic geometric principles to triangle area calculations without requiring advanced mathematical tools. The derivation is accessible to high school and undergraduate students, offering pedagogical value by demonstrating how fundamental geometric relationships lead to practical area computation methods. While mathematically equivalent to Heron's formula, this derivation pathway emphasizes geometric reasoning over algebraic manipulation.

Castel del Monte rappresenta uno dei più enigmatici esempi di architettura medievale, oggetto di numerosi tentativi interpretativi volti a individuare le regole geometriche sottese al suo tracciato planimetrico. La letteratura esistente si concentra prevalentemente su due filoni principali: lo schema basato sulla stella ottagona di Heinz Götze e l’impianto aureo-solare di Aldo Tavolaro, entrambi tuttavia distanti dai dati metrici rilevati con laser-scanner di ultima generazione. Attraverso un approccio innovativo, fondato sulla digitalizzazione delle misure e su algoritmi di minimizzazione, è stato individuato un modello geometrico inedito: un impianto rosoniforme costituito da 16 circoli tangenti disposti in tre corone concentriche, derivati dai basamenti dei torrioni. Il confronto con i rilievi dimostra scarti inferiori a 1 cm per la corte centrale e a meno di 3 cm per la muratura esterna, suggerendo una stretta corrispondenza con il tracciato originario. Questo risultato, verificato mediante un sistema analitico di classificazione basato su formulazioni algebriche, consente non solo di valutare criticamente le ipotesi storiche, ma anche di comparare e testare ogni possibile configurazione geometrica del monumento.

We propose a layered and component based software architecture style which supports interoperability in multiple databases (DB). The architectural style's building blocks and its constraints are described and the deployment of two design patterns outlined. Components placed in our architectural layers exhibit a linear topology and request/reply processing style. The constraints include communications between components which are not in the adjacent architectural layers and extension of the intuitive many : one bindings between components towards many : many. We comment on similarities with mediation architectures and outline some implementation issues

The article is devoted to the consideration of the issue of influential modifications of the genre system of modern mass media. It has been established that the mass media are one of the main means of communication for the wide audience. The meaning of the words "modification", "mass media", "mobile journalism", "new media" has been studied. The article notes that "new media" appeared in the 60s of the XX century. The main characteristics of the media are presented. It is noted that the most common examples of mass media are television, radio, newspapers, magazines, social media, digital media, the Internet. The textbooks and publications of domestic and foreign scientists, which raise the issue of influential modifications of the genre system of modern mass media are highlighted. It was found that the first media were newspaper, telegraph, radio, television. The study of the main types of media is emphasized: traditional media, prin...

he studio project in the first year of the undergraduate architecture program at the National School of
Architecture and Urbanism in Tunis provides an introduction to architectural design. It offers the learner the
opportunity to understand architectural space by perceiving, analyzing, designing, and finally transposing it
through drawing and various modes and tools of architectural representation and expression. In this article, we
present the methodological approach we have implemented to achieve one of the important didactic objectives:
the acquisition of architectural references and interdisciplinary knowledge related to the field of architecture.

Geometry has shown that the architecture was an expression of man and a basic principle that were always presented on an architectural work. The research was conducted through two steps with the first step was to identified the Tadao Ando’s geometry design principle through a content analysis approach. Based on the textual review, Tadao Ando's geometry design principle was tend to select the simple geometric forms as the embodiment of the building shapes. An user spatial experiences were created through inter space relationships and interactions with the surrounding environment through the geometry and shape processing and became the central to his architectural work creation. In other words, pure geometry concept was the instrument to presented all of those. The second step was to compared Tadao Ando's geometry design principles in the monistic architecture to the pluralistic architecture catagories using a precedent analysis approach. The comparisons toward the geometrical...

The existing building stock is recognised as a major contributor to total energy consumption and related carbon emissions around the globe. There is increased attention on the retrofit of existing building stock, especially residential buildings, as a way of curbing energy consumption and carbon emissions. Within this context, human nature connectedness (HNC) has the potential of further amplifying the benefits of sustainable buildings both from an energy conservation practice and tangible improvements to users’ satisfaction, health, and wellbeing. This study attempts to show a case study of the potential of using HNC through the adoption of biophilic design principles to improve a residential building performance. A terrace house located in Sydney, NSW, was used as a case study and proposed retrofit scenarios were simulated with DesignBuilder® and Rhinoceros/Grasshopper with a view of improved daylighting, thermal comfort, and energy consumption. The building performance is improve...

Review Article The topic of this article means "one can construct a circle (O, r), of which area is equal exactly to a given regular hexagon, using a straightedge and a compass only". No scientific theory lasts forever, but specific research and discoveries continuously build upon each other. The three classic ancient Greek mathematical challenges likely referring to are "Doubling The Circle", "Trisecting An Angle" and "Squaring The Circle", all famously proven Impossible under strict compass-and-straightedge constraints, by Pierre Wantzel (1837) using field theory and algebraic methods, then also by Ferdinand von Lindemann (1882) after proving π is transcendental. These original Greek challenges remain impossible under classical rules since their proofs rely on deep algebraic/transcendental properties settled in the 19th century. Recent claims may involve reinterpretations or unrelated advances but do overturn these conclusions above. Among these, the "Squaring The Circle" problem and related problems involving π have captivated both professional and amateur mathematicians for millennia. The title of this paper refers to the concept of "constructing a circle that has the exact area of a given regular hexagon," or "Circling The Regular Hexagon", for short. This research idea arose after the "Squaring The Circle" problem was studied and solved and published in "SJPMS" in 2024 [7]. This paper presents an exact solution to constructing a circle that is concentric with and has the same area as a given regular hexagon. The solution does not rely on the number π and adheres strictly to the constraints of Euclidean Geometry, using only a straightedge and compass. The technique of "ANALYSIS" is employed to solve this "Circling The Regular Hexagon" problem precisely and exactly with only a straightedge and compass, without altering any premise of the problem. This independent research demonstrates the solution to the challenge using only these tools. All mathematical tools and propositions in this solution are derived from Euclidean geometry. The methodology involves geometric methods to arrange the given circle and its equal-area regular hexagon into a concentric position. Building on this method of exact "Circling The Regular Hexagon," one can deduce an equivalent problem to formulate a new mathematical challenge: "Regular Heptagoning The Circle" (i.e., constructing a regular heptagon that has the same area as a given circle, using only a straightedge and compass). Keywords: Circling the regular hexagon, Circle; regular hexagoning circle; circle hexagonise; Circle area of an hexagon; constructing a circle with the same area as a hexagon; Euclidean geometry; straightedge and compass.

The topic of this article means "one can construct a regular hexagon of which area is equal exactly to a given circle (O, r), using a straightedge and a compass only". No scientific theory lasts forever, but specific research and discoveries continuously build upon each other. The three classic ancient Greek mathematical challenges likely referring to are "Doubling The Circle", "Trisecting An Angle" and "Squaring The Circle", all famously proven Impossible under strict compass-and-straightedge constraints, by Pierre Wantzel (1837) using field theory and algebraic methods [4], then also by Ferdinand von Lindemann (1882) after proving π is transcendental. These original Greek challenges remain impossible under classical rules since their proofs rely on deep algebraic/transcendental properties settled in the 19th century. Recent claims may involve reinterpretations or unrelated advances but do overturn these conclusions above. Among these, the "Squaring The Circle" problem and related problems involving π have captivated both professional and amateur mathematicians for millennia. The title of this paper refers to the concept of "constructing a regular hexagon that has the exact area of a given circle," or "Regular Hexagoning The Circle" for short. This research idea arose after the "Squaring The Circle" problem was studied and solved and published in "SJPMS" in 2024 [6]. This paper presents an exact solution to constructing a regular hexagon that is concentric with and has the same area as a given circle. The solution does not rely on the number π and adheres strictly to the constraints of Euclidean Geometry, using only a straightedge and compass. The technique of "ANALYSIS" is employed to solve the "Regular Hexagoning The Circle" problem precisely and exactly with only a straightedge and compass, without altering any premise of the problem. This independent research demonstrates the solution to the challenge using only these tools. All mathematical tools and propositions in this solution are derived from Euclidean geometry. The methodology involves geometric methods to arrange the given circle and its equal-area regular hexagon into a concentric position. Building on this method of exact "Regular Hexagoning The Circle," one can deduce an equivalent problem to formulate a new mathematical challenge: "Regular Heptagoning The Circle" (i.e., constructing a regular heptagon that has the same area as a given circle, using only a straightedge and compass). Keywords: Hexagoning the Circle; regular hexagoning circle; circle hexagonise; hexagon area of a circle; constructing a regular hexagon with the same area as a circle; Euclidean geometry; straightedge and compass.

This paper summarizes the ongoing research done on The Bishopsgate Tower in the City of London using parametric design methodologies. The process is indicative of how computational methods will develop in the future and help designers find solutions for increasingly complex spaces.

on the relationship between architecture and technology. Werner Sobek, whose practice operates at the interface between architecture and engineering, contributes with his expertise in lightweight constructions and his interest in performative structures. Complementary to these highly recognised professionals, Erik Demaine and Urs B. Roth contribute distinct perspectives on geometry. While they share a deep mathematical understanding and a passion for surprising "geometric discoveries" , their work methods are diametrically opposed. Erik Demaine develops his origami sculptures by engaging with computational algorithmic techniques, whereas Urs. B. Roth's heuristic search for form developed using sequences of accurate drawings and rigorous formulas that create their own peculiar aesthetic. Interesting enough, though, both consider themselves to be artists in addition to being scientists. The peer-review process for the selection of papers was managed by the scientific co-chairs, covering the domains of architecture, structural engineering, mathematics, and computer science, with the support of the scientific committee and the AAG2106 coordinator. From a very large pool of submissions, 22 papers have been accepted and included in the present proceedings. In addition, 25 posters have also been accepted for presentation, and the workshop chair has selected 12 workshop proposals for the pre-conference sessions. We extend our thanks and acknowledgements to all authors, tutors, reviewers, and organisers for their invaluable contributions to this process. External industry partners are vital to both the integrity and the execution of such a large conference. We are very pleased to have strong and highly supportive partners and industry sponsors for the AAG2016. We would like to especially thank our main sponsor ABB, our workshops sponsor Autodesk, and the conference sponsors Moog, Waagner Biro, ERNE, Evolute, Absolute Joint System, and Disney Research for their backing and guidance. Finally, we would also like to thank our colleagues and co-hosts at the Institute for Technology in Architecture (ITA), the Department of Architecture, and the ETH Zurich for their ongoing support and for providing the conference venues. We hope that you enjoy your time visiting ETH and Zurich, and wish you an excellent conference.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

Think a triangle is far from a circle? Our computational analysis reveals a surprising truth: an inscribed triangle already captures over 82% of a circle's circumference. This paper delves into the fascinating progression, demonstrating how the perimeter of a regular polygon swiftly and asymptotically converges to that of its encompassing circle as sides are added. Using a detailed computational model, we provide compelling numerical evidence of this geometric efficiency, charting the path from a mere 82.7% for a 3-sided polygon to near perfect alignment. We also precisely quantify edge cases, showing how a "2-sided polygon" fundamentally represents the circle's diameter, accounting for 31.83% of its circumference.

In recent years, substantial advancements in computer vision have been propelled by the advent of deep learning techniques. The transformative impact of deep learning is evident across various computer vision applications, spanning object detection, image classification, and semantic segmentation. Within the dynamic realm of architectural parametric design, a pivotal force shaping contemporary and future architectural pursuits, designers frequently encounter scenarios necessitating the reshaping, augmentation, or modification of building components and facades. This demand arises from diverse factors, including the integration of human-interactive facades, the implementation of sustainable facade systems, and more. In such instances, the foundational structure must possess malleability, enabling openness to edits. However, the grand scale and intricate details inherent in architectural designs pose a challenge, as a densely detailed mesh comprising millions of elements becomes increasingly unwieldy and challenging to modify for subsequent stages. This challenge underscores the critical need to generate objects capable of seamless evolution in tandem with evolving design requirements. This paper, will focus on categorizing 3D deep learning approaches into Learning-based 3D reconstruction in architecture and delving into their methods.

Quando sono arrivata in Italia, una delle prime cose che ho osservato, quasi senza volerlo, è stata la pianta delle città. Mi colpì subito la sua forma: un cerchio, poi un altro che lo circondava, e poi ancora un altro. Sembrava un disegno pensato non solo per organizzare lo spazio, ma per esprimere un'idea. Io l'ho letta così: una città costruita attorno a un centro, e da lì ogni abitante collocato a pari vicinanza, come se ognuno avesse diritto a essere vicino al cuore pulsante della comunità. Una disposizione che parla di equilibrio, di equità, una geometria che mi sembrò una testimonianza silenziosa ma potente di democrazia.

Clifford algebra or geometric algebra (GA) is a simple and intuitive way to model geometric objects and their transformations. Operating in high-dimensional vector spaces with significant computational costs, the practical use of GA requires dedicated software and/or hardware architectures to directly support Clifford data types and operators. In this paper, a family of embedded coprocessors for the native execution of GA operations is presented. The paper shows the evolution of the coprocessor family focusing on the latest two architectures that offer direct hardware support to up to five-dimensional Clifford operations. The proposed coprocessors exploit hardware-oriented representations of GA elements and operators properly conceived to obtain fast performing implementations. The coprocessor prototypes, implemented on field programmable gate arrays development boards, show significant speedups of about one order of magnitude with respect to the baseline software library Gaigen running on a general-purpose processor. The paper also presents an execution analysis of different GA-based applications, namely inverse kinematics of a robot, optical motion capture, raytracing, and medical image processing, showing good speedups with respect to the baseline general-purpose implementation.

This paper aims to, historically and empirically, study the moral panic and its relationship with emerging some modern phenomena in Iran’s society. The statistic sample included 185 individuals who were selected through stratified random sampling. Data were analyzed through religious commitment inventory (Worthingtonet et al., 2003), the questionnaire of socioeconomic base (Nabavi et al., 2008), and moral panic (Shahabi, 2008).  through descriptive statistics and regression analysis. Results showed that age and religiousness, having a positive significant relationship with moral panic, were the direct predictors, while the socioeconomic class was a reverse predictor of moral panic. In addition, findings showed that moral panic has been a main technique for competing with rivals. Moreover, considering the role of age and the increasing desire of the new generation towards modern phenomena, especially in the media field, it is recommended to recognize intergenerational distinctions and the media status in the development of community morality, and to provide appropriate cultural content for youth by educators.

This work proposes the use of partial order lattices along with representational schemes to account for patterns of ambiguity and emergence in the description of designs.The complexity of such designs is viewed as an aggregation of spatial layers that can all be decomposed by the subgroup relations of the symmetry of the configuration. At the end, this methodology points to a combinatorial approach that generates visual prototypes for future use in design synthesis. Here, Meier's work is just a case study that validates the group theoretical approach.

Review Article No scientific theory lasts forever, but specific research and discoveries continuously build upon each other. The three classic ancient Greek mathematical challenges likely referring to are "Doubling The Circle", "Trisecting An Angle" & "Squaring The Circle", all famously proven Impossible under strict compass-and-straightedge constraints, by Pierre Wantzel (1837) using field theory and algebraic methods, then also by Ferdinand von Lindemann (1882) after proving π is transcendental. These original Greek challenges remain impossible under classical rules since their proofs rely on deep algebraic/transcendental properties settled in the 19th century. Recent claims may involve reinterpretations or unrelated advances but do overturn these conclusions above. Among these, the "Squaring The Circle" problem and related problems involving π have captivated both professional and amateur mathematicians for millennia. The title of this paper refers to the concept of "constructing a circle that has the exact area of a given equilateral triangle " or "Circling The Triangle". This research idea arose after the "Squaring The Circle" problem was studied, solved and published in "SJPMS" in 2024 [8]. This paper presents an exact solution to constructing a circle that is concentric with and has the same area as any given regular triangle. The solution does not rely on the number π and adheres strictly to the constraints of Euclidean geometry, using only a straightedge and compass. The technique of "GEOMETRY ANALYSIS" is employed to solve this "Circling The Triangle" problem precisely and exactly with only a straightedge and compass, without altering any premise of the problem. This independent research demonstrates the solution to the challenge using only these tools. All mathematical tools and propositions in this solution are derived from Euclidean geometry. The methodology involves geometric methods to arrange the given regular triangle and its equal-area circle into a concentric position. Building on this method of exact "Circling The triangle," one can deduce an equivalent problem to formulate a new mathematical challenge: "Circling The Hexagon" (i.e., constructing a circle that has the same area as a given regular hexagon, using only a straightedge and compass).

Review Article No scientific theory lasts forever, but specific research and discoveries continuously build upon each other. The three classic ancient Greek mathematical challenges likely referring to are "Doubling the Circle", "Trisecting An Angle" & "Squaring The Circle", all famously proven Impossible under strict compass-and-straightedge constraints, by Pierre Wantzel (1837) using field theory and algebraic methods, then also by Ferdinand von Lindemann (1882) after proving π is transcendental. These original Greek challenges remain impossible under classical rules since their proofs rely on deep algebraic/transcendental properties settled in the 19th century. Recent claims may involve reinterpretations or unrelated advances but do overturn these conclusions above. Among these, the "Squaring the Circle" problem and related problems involving π have captivated both professional and amateur mathematicians for millennia. The title of this paper refers to the concept of "constructing a regular triangle that has the exact area of a given circle," or "Triangling the Circle". This research idea arose after the "Squaring the Circle" problem was studied and solved and published in "SJPMS" in 2024.This paper presents an exact solution to constructing a regular triangle that is concentric with and has the same area as any given circle. The solution does not rely on the number π and adheres strictly to the constraints of Euclidean geometry, using only a straightedge and compass. The technique of "ANALYSIS" is employed to solve the "Triangling the Circle" problem precisely and exactly with only a straightedge and compass, without altering any premise of the problem. This independent research demonstrates the solution to the challenge using only these tools. All mathematical tools and propositions in this solution are derived from Euclidean geometry. The methodology involves geometric methods to arrange the given circle and its equal-area regular triangle into a concentric position. Building on this method of exact "Triangling A Circle," one can deduce an equivalent problem to formulate a new mathematical challenge: "Hexagoning A Circle" (i.e., constructing a regular hexagon that has the same area as a given circle, using only a straightedge and compass).

developed a more detailed analysis in the second part of my PhD thesis [Cantù 2003, 153-345], where the accent was put on the discontinuities between the two editions, and on Grassmann's criticism of the 'traditional' definition of mathematics as a theory of magnitudes. 5 "I define as an extensive magnitude any expression that is derived from a system of units (none of which need to be the absolute unit) by numbers, and I call the numbers that belong to the units the derivation numbers of that magnitude; for example the polynomial a 1 e 1 + a 2 e 2 + … or Σ ae or Σ r r a e , where a 1 , a 2 , … are real

Dieses Buch beschreibt das architektonische Entwerfen ausgehend von seinen Medien und Werkzeugen. Die ersten Kapitel verschaffen einen Überblick über das heutige Wissen vom Entwerfen. Ist es ein schöpferischer Akt, ein kreativer Prozess oder ein Kreislauf immer wiederkehrender Handlungen? Als Basis für die weitere Untersuchung werden die wichtigsten Begriffe und Definitionen vorgestellt. Im Hauptteil des Buches geht es um das breite Spektrum von Medien und Werkzeugen, die zum Entwerfen genutzt werden. Deren Entstehung und spezifische Eigenschaften werden untersucht, um dann ihre Wirkungsweise zu analysieren und kritische Aussagen über ihren Gebrauch und ihre zukünftige Entwicklung zu treffen. Dies gilt besonders im Hinblick auf die Digitalisierung aller Entwurfswerkzeuge. Dieses Vorgehen erlaubt es, die vielfältigen Bedeutungsebenen architektonischer Handlungsmöglichkeiten zu erschließen.

Essay included in BS50: Business Standard celebration of India's 50th Anniversary of Independence.

The last sentence of the fifth paragraph should read as follows:
"Intimacy is (√5+1)/ 2:1" (the root went missing in the published piece).

Hence:
"Relationship is the only clue that we have to the world around us. The thingness of a thing is a chimera (behind every successful noun is a verb). Relationships are meaningful only when compared (fertilization is a comparison of chromosomes). Hence the proportion a:b as c:d. Or simpler still, a:b as b:c. If a is you, b is I and c is the other, then if I change so will the relationship between you and the other, between the whole Universe. And we can come closer still. Because a:b as b:a-&-b, or you:I as I:you-&-I. We have reached the ratio Phi of the Golden Rectangle. Intimacy is (√5+1)/ 2:1."

Note:
Fibonacci copied the Golden Ratio and many of his examples and algebraic techniques from the works of Abu Kamil (without crediting him).
Abu Kamil was known as Auoquamel in medieval Europe. His full name was Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ (c. 850 – c. 930) and lived in Egypt.

The author shows how to find both the equatorial circumference and polar circumference of the Earth in meters within the corner-socket, base perimeter dimensions and the height of the Great Pyramid, both of which produce the result of 365.242 inches, and 36,524.2 inches, expressing the days in one solar year and one century respectively.

This section introduces a selection of recent doctoral dissertations in architectural history that explore how architects have interpreted and engaged with the legacy of the past. From Renaissance models to modern and contemporary reinterpretations, the contributions critically examine the role of historical references in shaping design theories and practices. Emphasis is placed on methodological approaches, highlighting the relevance of historical awareness in architectural research and its role in addressing contemporary urban and cultural challenges.

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