The octant projection [1] [2] [3] or octants projection, is a type of map projection [4] proposed the first time, in 1508, by Leonardo da Vinci in his Codex Atlanticus. [5] Leonardo's authorship would be demonstrated by Christopher Tyler, who stated "For those projections dated later than 1508, [6] his drawings should be effectively considered the original precursors." [5] [7]
The same page of the Codex contains sketches of eight other projections of the globe (those known in the late fifteenth century) studied by Leonardo, including Ptolemy's conical planisphere projection and Roselli's pseudocylindric projection. [7]
The octant projection is the first known polyhedral map projection. It is neither conformal nor equal-area. [8] In it, the spherical surface of the earth is divided into eight octants, each flattened into the shape of a Reuleaux triangle bound by circular arcs. If transferred to an elastic support, it would be possible to cover with them the surface of a model of the earth's globe. [9] [10]
The eight triangles are oriented in a similar way as per two four-leaf clovers [11] side by side (fleurons), [12] being the earth poles in the center of each clove. One of the sides of the eight triangles, (the one opposite to the center of the pseudo clover), is one fourth of the equator, the remaining two (those that converge to the center of the pseudo clover), are part of the two meridians that with the equator dissect the globe in the eight octants. [13]
Projections also based on the Reuleaux triangle were published by:
Although Leonardo's first description of the octant projection has been proved by Tyler, [5] who decided to treat separately Leonardo's projection authorship (1508) from Leonardo's map authorship (1514), the other authors before him treat together the authorship of both map and projection, for they speak about "the eighth of a supposed globe represented in a plane" or about "globe sections" (Harrisse) or others about "gores", which are in fact a projection of the globe.
So, bearing in mind the fact that Tyler was the first scholar to mention the sketch of this projection in Codex Atlanticus in 2017, the authorship of the map it is not universally accepted, with some authors being completely against any minimal contribution from Leonardo, such as Henry Harrisse (1892), [15] or Eugène Müntz (1899 – citing Harrisse authority from 1892, although none of them talks about the projection's sketch in Codex Atlanticus). [16]
Other scholars accept explicitly both (map and projection: "the eight of a supposed globe represented in a plane"), completely as a Leonardo's work, describing the projection as the first of this type, among them, R. H. Major (1865) in his work Memoir on a mappemonde by Leonardo da Vinci, being the earliest map hitherto known containing the name of America, [17] Grothe, [18] the Enciclopedia universal ilustrada europeo-americana (1934), [9] Snyder in his book Flattening the Earth (1993), [10] Christopher Tyler in his paper (2014) "Leonardo da Vinci's World Map", [5] José Luis Espejo in his book (2012) Los mensajes ocultos de Leonardo Da Vinci, [19] or David Bower in his work (2012) "The unusual projection for one of John Dee's maps of 1580". [14]
Others also accept explicitly both (map and projection) as authentic, although leaving in the air Leonardo's direct hand, giving the authorship of the work to one of his disciples as Nordenskjold states in his book Facsimile-Atlas (1889) confirmed by Dutton (1995) and many others: "on account of the remarkable projection..not by Leonardo himself, but by some ignorant clerk", [20] or Keunig (1955) being more precise: "by one of his followers at his direction". [21]
The Codex Leicester is a collection of scientific writings by Leonardo da Vinci. The codex is named after Thomas Coke, Earl of Leicester, who purchased it in 1717. The codex provides an insight into the mind of the Renaissance artist, scientist and thinker, as well as an exceptional illustration of the link between art and science and the creativity of the scientific process.
A Reuleaux triangle is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circular disks, each having its center on the boundary of the other two. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because its width is constant, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?"
A world map is a map of most or all of the surface of Earth. World maps, because of their scale, must deal with the problem of projection. Maps rendered in two dimensions by necessity distort the display of the three-dimensional surface of the Earth. While this is true of any map, these distortions reach extremes in a world map. Many techniques have been developed to present world maps that address diverse technical and aesthetic goals.
Christopher William Tyler is a neuroscientist, creator of the autostereogram, and is the Head of the Brain Imaging Center at the Smith-Kettlewell Eye Research Institute He also holds a professorship at City University of London.
The Virgin and Child with Saint Anne is an unfinished oil painting by High Renaissance artist Leonardo da Vinci, dated to c. 1501–1519. It depicts Saint Anne, her daughter the Virgin Mary and the infant Jesus. Christ is shown grappling with a sacrificial lamb symbolizing his Passion as the Virgin tries to restrain him. The painting was commissioned as the high altarpiece for the Church of Santissima Annunziata in Florence and its theme had long preoccupied Leonardo.
The Madonna Litta is a late 15th-century painting, by Leonardo da Vinci, in the Hermitage Museum, Saint Petersburg. It depicts the Virgin Mary breastfeeding the Christ child, a devotional subject known as the Madonna lactans. The figures are set in a dark interior with two arched openings, as in Leonardo's earlier Madonna of the Carnation, and a mountainous landscape in aerial perspective can be seen beyond. In his left hand Christ holds a goldfinch, which is symbolic of his future Passion.
The Virgin and Child with Saint Anne and Saint John the Baptist, sometimes called the Burlington House Cartoon, is a drawing by Leonardo da Vinci. The drawing is in charcoal and black and white chalk, on eight sheets of paper that are glued together. Because of its large size and format the drawing is presumed to be a cartoon for a painting. No painting by Leonardo exists that is based directly on this cartoon, although the drawing may have been in preparation for a now lost or unexecuted painting commissioned by Louis XII. The drawing is the only extant larger-scale drawing by the artist.
The Chamberlin trimetric projection is a map projection where three points are fixed on the globe and the points on the sphere are mapped onto a plane by triangulation. It was developed in 1946 by Wellman Chamberlin for the National Geographic Society. Chamberlin was chief cartographer for the Society from 1964 to 1971. The projection's principal feature is that it compromises between distortions of area, direction, and distance. A Chamberlin trimetric map therefore gives an excellent overall sense of the region being mapped. Many National Geographic Society maps of single continents use this projection.
Leonardo da Vinci (1452–1519) was an Italian polymath, regarded as the epitome of the "Renaissance Man", displaying skills in numerous diverse areas of study. While most famous for his paintings such as the Mona Lisa and the Last Supper, Leonardo is also renowned in the fields of civil engineering, chemistry, geology, geometry, hydrodynamics, mathematics, mechanical engineering, optics, physics, pyrotechnics, and zoology.
Bernard Joseph Stanislaus Cahill, American cartographer and architect, was the inventor of the octahedral "Butterfly Map". An early proponent of the San Francisco Civic Center, he also designed hotels, factories and mausoleums like the Columbarium of San Francisco.
The Codex Atlanticus is a 12-volume, bound set of drawings and writings by Leonardo da Vinci, the largest single set. Its name indicates the large paper used to preserve original Leonardo notebook pages, which was used for atlases. It comprises 1,119 leaves dating from 1478 to 1519, the contents covering a great variety of subjects, from flight to weaponry to musical instruments and from mathematics to botany. This codex was gathered in the late 16th century by the sculptor Pompeo Leoni, who dismembered some of Leonardo's notebooks in its formation. It is now in the Biblioteca Ambrosiana in Milan.
The Waterman "Butterfly" World Map is a map projection created by Steve Waterman. Waterman first published a map in this arrangement in 1996. The arrangement is an unfolding of a polyhedral globe with the shape of a truncated octahedron, evoking the butterfly map principle first developed by Bernard J.S. Cahill (1866–1944) in 1909. Cahill and Waterman maps can be shown in various profiles, typically linked at the north Pacific or north Atlantic oceans.
The Madrid Codices I–II, are two manuscripts by Leonardo da Vinci which were discovered in the Biblioteca Nacional de España in Madrid in 1965 by Dr. Jules Piccus, Language Professor at the University of Massachusetts. The Madrid Codices I was finished during 1490 and 1499, and II from 1503 to 1505.
The Cahill–Keyes projection is a polyhedral compromise map projection first proposed by Gene Keyes in 1975. The projection is a refinement of an earlier 1909 projection by Bernard Cahill. The projection was designed to achieve a number of desirable characteristics, namely symmetry of component maps (octants), scalability allowing the map to continue to work well even at high resolution, uniformity of geocells, metric-based joining edges, minimized distortion compared to a globe, and an easily understood orientation to enhance general usability and teachability.
Leonardo's world map is the name assigned to a unique world map drawn using the "octant projection" and found loosely inserted among a Codex of Leonardo da Vinci preserved in Windsor. It features an early use of the toponym America and incorporates information from the travels of Amerigo Vespucci, published in 1503 and 1505. Additionally, the map depicts the Arctic as an ocean and Antarctica as a continent of about the correct size.
An overlapping circles grid is a geometric pattern of repeating, overlapping circles of an equal radius in two-dimensional space. Commonly, designs are based on circles centered on triangles or on the square lattice pattern of points.
Augusto Marinoni was professor of romance philology at the Università Cattolica del Sacro Cuore of Milan, a member of the Commissione Vinciana and the Accademia dei Lincei. He is considered one of the greatest scholars of Leonardo da Vinci.
A polyhedral map projection is a map projection based on a spherical polyhedron. Typically, the polyhedron is overlaid on the globe, and each face of the polyhedron is transformed to a polygon or other shape in the plane. The best-known polyhedral map projection is Buckminster Fuller's Dymaxion map. When the spherical polyhedron faces are transformed to the faces of an ordinary polyhedron instead of laid flat in a plane, the result is a polyhedral globe.
José Luis Espejo Pérez, is a Spanish-language writer specializing in historical essays.