Tessellation is “a collection of shapes [tiles] that fit together without gaps or overlap to cover the infinite mathematical plane” (Fathauer, 2021). Most tilings are “periodic,” in the sense that the pattern repeats itself when “translated” (shifted without rotation). In the 1970s Roger Penrose described several sets of tiles that could cover the plane aperiodically. The search then began for the “einstein” (one stone) – a single tile that could cover the plane aperiodically. In March of 2023, Smith, Myers, Kaplan & Goodman-Strauss described a tile, commonly known as the “hat” that covered the plane aperiodically. However, to do so, this tile had to be occasionally turned over (to make its mirror image). Subsequently in May of 2023, the same authors reported another tile that could cover the plane aperiodically without any need for mirror images. This tile was called the “spectre.” This posting briefly reviews these recent developments in a style more visual than verbal.
Archive for Architecture
This post describes some of the events that occurred in Ronda, a town in southern Spain, during the summer of 1936. After the Spanish Civil War broke out, Anarchists quickly took control of the town, and murdered many supporters of the Nationalist cause. Two months later, advancing Nationalist forces captured Ronda, and drove most of its people from their homes. Those that refused to leave suffered bloody reprisals. These events quickly became mythic rather than historic. In one story, the Anarchists had murdered the town’s Falangists by having them beaten to death in the town’s plaza and then thrown into the canyon that cuts through the center of the town. Ernest Hemingway recounted this version in his 1940 novel For Whom the Bell Tolls. However, most historians now agree that this never happened.
From the 12th to 15th Centuries groups of people called the Cathars lived quietly in various regions of Western Europe – Northern Italy, the Rhineland and, most especially, Southern France. They followed the moral teachings of Jesus, forsaking worldly goods and loving one another, but they did not believe in the basic theology of Christianity. They considered that the world was evil, that human beings were spirits imprisoned in the flesh, and that the soul could only be set free at death if one had lived a life of purity. The Catholic Church considered these beliefs heretical, and in 1208 Pope Innocent III called for a crusade to eradicate the heresy. Named after the inhabitants of the city of Albi which had a flourishing Cathar population, the Albigensian Crusade lasted from 1209 until 1229. After years of terrible violence and cruelty, most of those who professed Cathar beliefs were dead. All that now remains of these peaceful people are the ruins of the hilltop castles in which they sought refuge.
In 1098 a small group of monks left the Benedictine monastery of Molesme in Burgundy to live in the forest of Cîteaux (Latin Cistercium) just south of Dijon. They considered their original home too lax and luxurious, and wished to return to the austere life of solitude, chastity, poverty and manual labor that St Benedict had originally proposed in the 6th Century. They distinguished themselves from the Benedictines by wearing undyed white robes rather than black. Over the next hundred and fifty years, the small monastery founded by these “white monks” at Cîteaux became the center of the Cistercian Order, which linked together over 500 abbeys in Europe, extending from Portugal to Estonia and from Sicily to Norway. The Cistercians were noted for the proficiency of their agriculture, the fervor of their scholarship, and the beauty of their buildings. This post comments on their achievements.
The Disintegration of the Roman Empire
Constantine the Great (reign 306-337) re-unified the Roman Empire and promoted Christianity as the imperial religion. For several decades after Constantine, the Empire remained stable. However, after the death of Theodosius I (reigned 379-395), the Empire fractured into Eastern and Western regions, each ruled by one of his sons.