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I think it's the most beautiful dome in the worls and its building history is simply extraordinary

From arch.mcgill.ca:

Architect, perspectivist, and sculptor. There’s not much information about Filippo Brunelleschi's career until 1418-20, when we he was engaged in various architectural projects for Florence: the dome of Santa Maria del Fiore, the Spedale degli Innocenti, and the churches of San Lorenzo and Santo Spirito. In this period he painted, by the aid of a mirror, two panels showing the Baptistery and the Piazza della Signori perspective. These works - now lost - marked the birth of linear perspective. Also in the same years, Brunelleschi built mechanical clocks, of which only indirect accounts survive. His work as sculptor is better documented. In fact, it was back in a competence in 1401 that he competed unsuccessfully against Lorenzo Ghiberti for the construction of the second bronze door of the Florence Baptistery.
In 1418 Brunelleschi entered a new competition against Ghiberti to submit a model for the dome of Santa Maria del Fiore. Brunelleschi won the contest in 1420 with a proposal to erect the dome without wooden centering.
The construction of the dome began in the summer of 1420 and was finished by 1436. The lantern, designed by Brunelleschi, was completed after his death. The copper sphere was placed on the top by Verrocchio in 1472. The construction embodies a rigorously geometrical design.

The octagonal drum carries eight "shells." Empty spaces have been left inside each of these to lighten the massive structure. The angle ribs are not intended to provide structural support. Another notable feature is the external covering system, which consists of tiles specially designed for easy assembly and maintenance.

Dimensions and characteristics
The dome's springing point stands 177 feet above ground level, while its height from the drum base to the top is about 108 feet. The distance between two opposite edges of the exterior octagonal base is about 176 feet. The height of the lantern atop the dome is slightly more than 72 feet. The dome weighs an estimated 37,000 metric tons, and the number of bricks used in the structure may exceed four million.



Geometrical structure of the dome
Polygonal domes, such as the eight-sided dome of Santa Maria del Fiore, are more complex in their design than round domes, which are generated by rotating a quarter-circle around a vertical axis. To build a dome without the use of a supporting framework, each of the masonry rings that compose the dome must be completed in succession. This was the method used by Brunelleschi, and illustrated in the model of the masonry layers.
The bricks were laid on sloping beds. Before closing each ring of bricks, the workmen placed a row of bricks whose longer sides protruded with respect to the bricks resting on the conic surface. This arrangement, known as a herring-bone, displays a spiral profile.
The second model shows the geometric principles of Brunelleschi's dome and illustrates two other essential features. First, we can see the method used to obtain the so-called pointed-fifth curvature of the angle ribs. Brunelleschi took the circle in which the dome's inner octagon is inscribed, and divided it into five equal parts. He traced intersecting arcs with a compass opening equal to four-fifths of the circle's diameter.
Second, the model shows the characteristic profile of the brick beds, known as "slack line" because it resembles a loose string. The reason for this is that the brick beds all lie on the surface of an inverted cone whose axis coincides with that of the dome. The cone's vertex shifts upwards as the work proceeds. This construction method is comparable to what happens when we use a pencil sharpener. The polygonal pencil is the equivalent of the dome, while the cone-shaped opening of the device represents the dome's inverted cone design. The sharpener turns the pencil's vertical sides into conic surfaces.