Union of Hypercubes and 3D Minkowski Sums with Random Sizes

Let T={triangle_1,...,triangle_n} be a set of of n pairwise-disjoint triangles in R^3, and let B be a convex polytope in R^3 with a constant number of faces. For each i, let C_i = triangle_i oplus r_i B denote the Minkowski sum of triangle_i with a copy of B scaled by r_i>0. We show that if the scaling factors r_1, ..., r_n are chosen randomly then the expected complexity of the union of C_1, ..., C_n is O(n^{2+epsilon), for any epsilon > 0; the constant of proportionality depends on epsilon and the complexity of B. The worst-case bound can be Theta(n^3). We also consider a special case of this problem in which T is a set of points in R^3 and B is a unit cube in R^3, i.e., each C_i is a cube of side-length 2r_i. We show that if the scaling factors are chosen randomly then the expected complexity of the union of the cubes is O(n log^2 n), and it improves to O(n log n) if the scaling factors are chosen randomly from a "well-behaved" probability density function (pdf). We also extend the latter results to higher dimensions. For any fixed odd value of d, we show that the expected complexity of the union of the hypercubes is O(n^floor[d/2] log n) and the bound improves to O(n^floor[d/2]) if the scaling factors are chosen from a "well-behaved" pdf. The worst-case bounds are Theta(n^2) in R^3, and Theta(n^{ceil[d/2]}) in higher dimensions

Procedural Generation of Voxel Worlds with Castles

This thesis explores procedurally creating voxel based terrains and creating castles in them. With the explosion of interest in using voxels in games since the success of Minecraft, and even big budget titles like EverQuest Next using this technology, this research aims to address an area that has not received adequate attention: procedural generation of buildings like castles. By utilizing voxels combined with procedural generation of terrain, games like Minecraft and its many clones are able to offer worlds of near limitless size for the player to explore. However, this presents a challenge: how to fill a world with content worth exploring; primarily structures that look human-made. This project focuses on perhaps the most iconic of fantasy game structures: the castle. A voxel world engine was developed to explore this problem, complete with a procedural terrain generator similar to that used in voxel world games. The terrain generator developed for this project is capable of creating a variety of terrains randomly to test castle placement and generation. By drawing inspiration from real castles construction algorithms were developed to analyze areas of voxel terrain for good castle positioning and adaptive layout of a castle to the terrain. This was used to procedurally generate castles that aim to look as though they were placed into the world by a human agent

An Investigation of the Duality Between Art and Math

The following paper is an investigation of the relationship that exists between math and art. It argues the importance of integration between the two disciplines by shedding light on unrecognized characteristics within both. The paper provides examples of mathematical art and beautiful proofs which help unearth the potential of mixing math and art at an academic level

Geometry in early muslim architecture

Proportional systems in the Muslim decorative arts are often manifest and easier to detect than those which may underlie their architectural forms. In order to study the plans of early Muslim architecture and gain understanding of the method by which they were generated, fourteen examples of early Muslim buildings, dating from AD 692 to 1125, have been geometrically analysed using measured plans and actual dimensions. These examples cover a vast area of the Muslim world, from Syria and Iraq in the north, to Egypt and Córdoba in the west. The results of the analysis demonstrate that a proportional system, familiar to most ancient cultures and based on a circle inscribed within a square and subdivided to create a set of grid lines, relating in a proportion of 1:2 and 1: 12, proves to be of notable relevanceThe thesis initially explores the nature of the evidence and the limits of inference in relation to the subject of geometry in early Muslim architecture, then focuses on the architectural planning of the Near East prior to the coming of Islam and on the work of medieval Muslim scholars. The thesis examines the concept of squaring the circle, its components, and its role as a proportional system in two -dimensional Islamic architectural decoration.Chapter Four ( "Geometric Analysis of Individual Buildings "), is the main part of the thesis, setting out an investigation of the proportional system of structures varying from mosques and palaces to cupolas and mihrabs. It is intended to be read in conjunction with vol. II of the thesis, which contains the analytical drawings and proposed geometric schemes.By pursuing two separate lines of enquiry on each of the selected buildings, the first based on geometrical lines superimposed on published plans, the second through mathematical calculation using actual dimensions, it is proved beyond doubt that early Muslim buildings were in fact geometrically organized.Finally, the thesis concludes by considering alternative systems of analysis in order to compare its findings and establish the validity of its proposed geometric schemes for the examples selected

Optimizing Communication in Twenty-First Century Residential Architecture in Hawai‘i.

D.Arch. Thesis. University of Hawaiʻi at Mānoa 2018

THE MAIN RESULTS OF ARCHAEOLOGICAL RESEARCH AT REȘCA-ROMULA (1869-2019) AND CONSIDERATIONS ON THE GEOPHYSICAL APPROACH – PART II

Book Chapter in INSIGHTS OF GEOSCIENCES FOR NATURAL HAZARDS AND CULTURAL HERITAGE, Editor: Florina CHITE

Cultural Considerations: Ecolodge in Mexico

A trend of extensive globalization and homogenization of architecture and culture is occurring. As globalization increases cultural identity in architecture becomes more precious; the need to create a distinctive sense of place becomes more desirable. Investigative research on cultural considerations will create a deeper understanding of a people and their culture. This research explores the indigenous roots of the Chihuahua, Mexico responding to aspects of physical constitutes, sustainability, history, culture, environmental policy and adventure tourism. First, the research investigates geographical and environmental components that immediately influence the region and analyze how these qualities make the site unique. Next, the research studies the history of this indigenous community, from its beginnings of a modern civilization to the current state of being. Similarities found in past and present can give core relations to how buildings respond environmentally and socially to use. Insight gained from research helps to better understand the needs and lifestyles of the Chihuahuan people. The dynamics of these users reveal how architectural form can better accommodate their social interaction. Finally, the research promotes and supports the design of a boutique eco-hotel within the rural community