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 Painting
Matsuo Basho (1644-1694), one of the most famous poets of Japan, was a master of the haiku, a poetic form in which an abundance of meaning is concentrated into a paucity of syllables. Basho travelled widely in Japan, writing about t his experiences in a fascinating mixture of prose and poetry. In 1689 he undertook his longest journey: from Edo into the far north of Japan, a region known as Oku. His record of that journey is known as Oku no Hosomichi (The Narrow Road to the North).
In 1891, Paul Gauguin (1848-1903) left his wife and five children and sailed for Tahiti, where he hoped
to immerse myself in virgin nature, to see no one but savages, live their life, with no other thought in mind but to render, the way a child would, the concepts formed in my brain and to do this with the aid of nothing but the primitive means of art, the only means that are good and true (letter quoted in Eisenman, 1997, p 77).
His decision to desert his family and follow his art has been considered by philosophers as a case study in ethics. Was his hope of artistic success adequate justification for his behavior? As luck would have it, Gauguin did become a famous artist, albeit posthumously. Can this retrospectively vindicate his flight to Tahiti? These issues are complex – both in the abstract and in terms of Gauguin’s actual life.
In 1891 Mary Cassatt (1844-1926) exhibited a set of ten color prints at the Durand-Ruel Gallery in Paris. These prints had been made using new aquatint procedures that allowed the artist to print solid blocks of color. The colors and patterns owed much to the Japanese woodblock prints which had recently been exhibited at the Ecole des Beaux Arts. Another source of the imagery was the work of Edgar Degas (1834-1917), with whom Cassatt had previously worked on black and white etchings and aquatints. Though printed in a small edition size of 25, Cassatt’s 1891 prints did not sell well, perhaps because they were too innovative for the market. Now they are appreciated as key contributions to the art of the modern print.
Edward Hopper (1882-1967) painted the independence and the loneliness of 20th-Century America. He was a realist in the days when most painters tended toward the abstract. Yet his paintings incite the imagination far more than the works of any abstract expressionist. His enigmatic images force the viewer to wonder what is going on:
Hopper was neither an illustrator nor a narrative painter. His paintings don’t tell stories. What they do is suggest—powerfully, irresistibly—that there are stories within them, waiting to be told. He shows us a moment in time, arrayed on a canvas; there’s clearly a past and a future, but it’s our task to find it for ourselves. (Block, 2016, p viii).
More than any other painter, Hopper has inspired writers to find the stories and meanings behind his paintings. This post summarizes his life, describes his working methods, and presents some of his pictures together with the writings they have stimulated.
Paul Cézanne (1839-1906) learned to experience nature with the vividness of the Impressionists but evolved his own individual style of painting. How he perceived the world was as important as the way it appeared. For most of his life he lived and painted in Aix-en-Provence. He had no students and his work became recognized only toward the end of his life. Nevertheless, many of the proponents of the modernist movement that began in the first decade of the 20th Century acknowledged Cézanne as their artistic father (Hook, 2021). This post comments on some of his paintings.
In the gospels of the Christian New Testament, Mary Magdalene was the first person to recognize the risen Christ. He told her to tell the disciples the news of his resurrection, thus honoring her as the “apostle to the apostles.” In the Gnostic Gospels she appears as a visionary disciple of Jesus. In the centuries after her life, her story was conflated with that of the sinful woman who anointed the feet of Jesus at a feast in the house of Simon, and Mary thus became a model of repentance. This posting discusses these and other ways in which we conceive of Mary Magdalene.
Religious belief differs from everyday belief. Since it cannot be tested or independently confirmed, religious belief must be accepted on faith. Religious belief generally starts with a few powerful and attractive ideas. For example: would it not be wonderful if we did not have to die? As time passes these foundational principles are elaborated and bolstered by other equally untestable beliefs to form a relatively coherent set of teachings. These “doctrines” can then organize communities of the faithful, govern the behavior of believers, and attract new converts. Some believers may choose to interpret the foundational ideas of a faith differently from the system of beliefs that are considered “orthodox” (Greek: ortho straight, correct + doxa, opinion). Beliefs that differ from the orthodox are termed heretical (Greek: hairesis, choice). Heresies are usually considered dangerous since they can easily disrupt the accepted doctrine and question the power of those who promote orthodoxy. Heresy occurs in the history of all the world’s religions (Henderson, 1988) This post limits itself to the early Christian beliefs and heresies about the nature of God, particularly those concerning the Trinity.
Frames of Reference: The Art of William Kurelek
This post discusses the life and work of William Kurelek (1927-1977), one of the most distinctive and prolific Canadian painters of the latter half of the 20th Century. Kurelek was a figurative artist during the heyday of Abstract Expressionism, and a fervent Christian artist in the years of unbridled secularism. His work should be considered in the context of a life framed by memory, madness and religion (Kurelek, 1980; Morley, 1986). The post contains illustrations of many of his paintings, which can speak for themselves independently of my commentary.
All the major religions of the present world are androcentric in nature and misogynistic in practice. The following are some typical injunctions in the Christian scriptures:
Let your women keep silence in the churches: for it is not permitted unto them to speak; but they are commanded to be under obedience as also saith the law.
And if they will learn any thing, let them ask their husbands at home: for it is a shame for women to speak in the church. (I Corinthians 14: 34-35)
Let the woman learn in silence with all subjection.
But I suffer not a woman to teach, nor to usurp authority over the man, but to be in silence. (1 Timothy 2: 11-12)
These rulings are in spite of (or perhaps because of) women being more attentive to religious teachings, and participating more often in religious services than men (Pew Research Foundation, 2016). The two passages nevertheless serve a purpose – they provide clear evidence that the New Testament does not always represent the word of God.
The androcentricity of organized religion differs completely from prehistoric religious beliefs, wherein God was more likely female than male (Stone, 1978). Over recent centuries, however, female aspects of the godhead have become more and more recognized. This posting briefly considers some of the manifestations of the divine feminine, and mentions what might be involved in a feminist theology.