A solar “eight” in the Cathedral of Palma

Source: Wikimedia Commons

Source: Wikimedia Commons

Palma’s cathedral, also known as La Seu, like almost every building of this kind hide curiosities and mathematical facts that surprise due to the construction’s age. We can find trigonometry relations in a common cathedral or church, some of them even with a religious meaning behind. But in La Seu we can find two particular incidents that catch our eye. The first one happens every 2nd of February (2/2) and 11th of November (11/11), when the sunbeams go throw the east-faced rose window (known as Oculus Maior) and they are projected under the west-faced rose window, tangentially, in such a way that their centers lie on a line that is perpendicular to the ground.

Sorice: XEIX webpage

Source: XEIX webpage

It is not a coincidence the days in which the event takes place, as they are both in a similar position regarding the winter solstice. Thanks to this light  effect and the data currently available on the internet, we can easily calculate La Seu’s orientation. When this phenomenon takes place, the direction of the sunbeams coincide with the nave’s orientation. Then, we only have to set the time and figure out the exact azimuth considering this time and geographic situation. With some of the available programs, we can find out that it has a value of 122,4º and the angle of solar elevation is 10,2º (both calculus with an error smaller than 0,5º!).

Source: XEIX webpage

Source: XEIX webpage

The other incident takes place during the days near to the winter solstice. We can stare at the sunrise going through both rose windows causing a rather impressive light effect. Then again, we can find the value of the azimuth which results in 120,3º. Curiously, the bell tower, which has a square base, doesn’t have an axis of symmetry parallel to the central nave, they are out of place about 10º from each other.

Source: XEIX webpage

Source: XEIX webpage

Like many others cathedrals and churches, this one is built on an ancient mosque. In this culture, it was very important to have the ”qibla” oriented towards Mecca, specifically, towards the Ka’ba, following the precepts of the Koran. The muslim domination in Mallorca happened between 903 and 1231. In this period of time, the solution of the “qibla problem” was already known, solved by al-Khwârizmî in the 9th century. In effect, if one traces the line segment bisector of the east-faced bell tower and lengths it over the terrestrial sphere it matches the Ka’ba with an astonishing precision.

You can read more information about these two facts in this interesting article.

This post has been written by Aina Ferra in the subject Història de les Matemàtiques (History of Mathematics, 2014-15)

Location: Cathedral of Palma (map)

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