The Doni Tondo or Doni Madonna, is the only finished panel painting by the mature Michelangelo to survive. Now in the Uffizi in Florence and still in its original frame, the Doni Tondo was commissioned by Agnolo Doni to commemorate his marriage to Maddalena Strozzi, the daughter of a powerful Tuscan family; the painting is in the form of a tondo, meaning in Italian,'round', a shape, associated during the Renaissance with domestic ideas. The work was most created during the period after Doni's marriage in 1503 or 1504, before the Sistine Chapel ceiling frescoes were begun in 1508; the Doni Tondo portrays the Holy Family in the foreground, along with John the Baptist in the middle-ground, contains five nude male figures in the background. The inclusion of these nude figures has been interpreted in a variety of ways. Mary is the most prominent figure in the composition, taking up much of the center of the image, she sits directly on the ground without a cushion between herself and the grass, to better communicate the theme of her relationship to the earth.

Joseph is positioned higher in the image than Mary, although this is an unusual feature in compositions of the Holy Family. Mary is seated between his legs, as if he is protecting her, his great legs forming a kind of de facto throne. There is some debate as to whether Mary is receiving the Child from vice versa. Saint John the Baptist, the patron saint of Florence, is commonly included in Florentine works depicting the Madonna and Child, he is in the middle-ground between the Holy Family and the background. The scene appears to be a rural one, with the Holy Family enjoying themselves on the grass and separated from the curiously unrelated group at the back by a low wall; the painting is still in its original frame, one that Michelangelo might have influenced or helped design. The frame is ornately carved and rather unusual for the five heads it contains which protrude three-dimensionally into space. Similar to the nudes of the background, the meanings of these heads has been the subject of speculation.

The frame contains carvings of crescent moons, stars and lions’ heads. These symbols are references to the Doni and Strozzi families, taken from each one's coat of arms; as depicted on the frame, “the moons are bound together with ribbons that interlock with the lions,” referring to the marriage of the two families. There is a horizontal band a wall, separating the foreground and background; the background figures are five nudes, whose meaning and function are subject to much speculation and debate. Because they are much closer to us, the viewers, the Holy Family is much larger than the nudes in the background, a device to aid the illusion of deep space in a two-dimensional image. Behind Saint John the Baptist is a semi-circular ridge, against which the'ignudi' are leaning, or upon which they are sitting; this semi-circle reflects or mirrors the circular shape of the painting itself and acts as a foil to the vertical nature of the principal group. Mary and Joseph gaze at Christ; the far background contains a mountainous landscape rendered in atmospheric perspective.

The Doni Tondo is believed to be the only existing panel picture Michelangelo painted without the aid of assistants. The juxtaposition of bright colors foreshadows the same use of color in Michelangelo's Sistine Ceiling frescoes; the folds of the drapery are modelled, the modelling of the figures is distinctly sculptural, suggesting they are carved in medium marble. The nude figures in the background have softer modelling and look to be precursors to the ignudi, the male nude figures in the Sistine Ceiling frescoes. Michelangelo's technique includes shading from the most intense colors first to the lighter shades on top, using the darker colors as shadows. By applying the pigment in a certain way, Michelangelo created an "unfocused" effect in the background and focused detail in the foreground; the most vibrant color is located within the Virgin's garments, signifying her importance within the image. The masculinity of Mary could be explained by Michelangelo's use of male models for female figures, as was done for the Sistine Chapel.

Michelangelo used a limited palette of pigments comprising Lead White, Verdigris and a few others. He avoided ochres and used little vermilion; the composition is, most partially influenced by the cartoon for Leonardo da Vinci’s The Virgin and Child with St. Anne. Michelangelo's Holy Family forms a tight, separated group in the centre foreground of the image, with the Virgin's figure constructing a typical Renaissance pyramid or triangle. Michelangelo saw the drawing in 1501 while in Florence working on the David; the Doni Tondo is associated with Luca Signorelli’s Medici Madonna in the Uffizi. Michelangelo knew of the work and its ideas, he wanted to incorporate those ideas into his own work. Signorelli's Madonna uses a tondo form, depicts nude male figures in the background, displays the Virgin sitting directly on the earth. Three aspects of the painting can be attributed to an antique sardonyx cameo and a 15th-century relief from the circle of Donatello, available to Michelangelo in the Palazzo Medici: the circular form, the masculinit

In mathematics, a differentiable manifold M of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at every point p of M the tangent vectors provide a basis of the tangent space at p. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section on M. A particular choice of such a basis of vector fields on M is called a parallelization of M. An example with n = 1 is the circle: we can take V1 to be the unit tangent vector field, say pointing in the anti-clockwise direction; the torus of dimension n is parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take n = 2, construct a torus from a square of graph paper with opposite edges glued together, to get an idea of the two tangent directions at each point. More every Lie group G is parallelizable, since a basis for the tangent space at the identity element can be moved around by the action of the translation group of G on G.

A classical problem was to determine. The zero-dimensional case S0 is trivially parallelizable; the case S1 is the circle, parallelizable as has been explained. The hairy ball theorem shows; however S3 is parallelizable, since it is the Lie group SU. The only other parallelizable sphere is S7; the parallelizable spheres correspond to elements of unit norm in the normed division algebras of the real numbers, complex numbers and octonions, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, requires algebraic topology; the product of parallelizable manifolds is parallelizable. Every orientable three-dimensional manifold is parallelizable. Any parallelizable manifold is orientable; the term framed manifold is most applied to an embedded manifold with a given trivialisation of the normal bundle, for an abstract manifold with a given stable trivialisation of the tangent bundle. A related notion is the concept of a π-manifold. A smooth manifold M is called a π-manifold if, when embedded in a high dimensional euclidean space, its normal bundle is trivial.

In particular, every parallelizable manifold is a π-manifold. Chart Differentiable manifold Frame bundle Kervaire invariant Orthonormal frame bundle Principal bundle Connection G-structure Bishop, R. L.. I. Tensor Analysis on Manifolds, The Macmillan Company, ISBN 0-486-64039-6 Milnor, J. W.. D. Characteristic Classes, Princeton University Press Milnor, J. W. Differentiable manifolds which are homotopy spheres, mimeographed notes

Dwight B. Heard was a rancher in Arizona, along with the president of the Arizona Cotton Association, he is famous for publishing the Arizona Republican, now The Arizona Republic, from 1912 to 1929. He was a delegate to the Republican National Convention in 1928, he died in 1929, a few months before the Heard Museum, a Native American art museum named after him was opened. Heard moved to Chicago from Wayland, shortly after high school, he began working at Hibbard, Spencer and Company. During his time as an employee, Heard met his wife, Maie Bartlett, while being mentored by Adolphus Bartlett, the father of Maie. In 1893, they were married. Just one year the couple moved to Arizona after Heard was diagnosed with lung ailments, they decided to make it their home. In Arizona, Heard was one of the largest landowners in the Salt River Valley, he owned the Bartlett-Heard Land and Cattle Company, which sold cattle, citrus trees and cotton in South Phoenix. He was the president of the Arizona Cotton Growers' Association, was credited for making Arizona's cotton industry more competitive.

His other business interests included real estate investment lending. He was a delegate to the Republican National Convention in 1928. In 1912 Heard purchased the Arizona Republican, now the Arizona Republic, published it until his death in 1929. Soon after his death, the Heard Museum was founded, housing Native American artifacts the Heards had acquired during their life in Phoenix. Maie Heard worked as the director of the museum for twenty years, she died 22 years after her husband Dwight's death, on March 14, 1951. Heard Museum website