# A solar “eight” in the Cathedral of Palma

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.

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

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

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.

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)

Photography by Laura Barber

The name of the Talaiotic culture comes from conic towers built with stones, probably used as a dwelling, watch towers and defense towers. These tables (“Taules” in Catalan) consist in a vertical rectangular stone and another one placed horizontally on its top, so the name of the table is given by the form of “T”. But… why these old monuments are mathematic? The front of most of them is oriented to the south! This orientation is related to the possible use as calendar in this former culture. The construction of the first monuments in Balearic islands dates from the end of the 2nd millenium BC to the beginnings of the 1st millenium BC. At this moment, these monuments began to proliferate on Mallorca and Menorca (there are 31 only in this small Mediterranean island!) appearing in isolated fashion as a territorial boundary stone.

The tables served as sanctuaries next to other monuments and all of them were built in almost the same latitude (and longitude?). For example, Sa naveta des Turons (latitude = 39.99º and longitude = 3.93º), Torretrencada (latitude = 40.003º and longitude = 3.89º) and Torre d’en Gaumès (latitude = 39.93º and longitude = 4.12º) seems to be aligned!

Naveta des Tudons
Source: Wikimedia Commons

In 1996, Vicente Ibáñez Orts published his hypothesis on the Table explaining that their design was very well computed and not the result of chance. Regarding Torretrencada, it seems that the monument was built from some mathematical computation indicating that Talaiotic men had a system of writing numbers and a deep knowledge of arithmetic and geometry)

This post has been written by Laura Barber and Anabel Luís in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).

# Kukulkan’s temple in Chichen Itza

Kukunkan’s temple
Photography by Roberto Lara

Today, the ancient Mayans are particularly famous by their incredible calendar. In fact, Mayans made a really powerful calendar inspired by astronomical events, as they really were essentially farmers and very superstitious. This is the reason why they didn’t have an unique counting system in their calendar, that is, they had ‘sub-calendars’ which different periods as reference. For example, they had a holy calendar (called Tzolkin), which had 260 days, and also a civil solar calendar (called Haab) with 365 days (it’s not clear what was the motivation for the Tzolkin). Tzolkin means “division of days” was probably based on the 224-day Venus sidereal period although there are some hypothesis which defend that it is related with the human gestation period. The Haab calendar consisted in 18 months of 20 days each plus an additional period of five days at the end of the year. It was first used around 500 BC. Mayans were so religious and these astronomical calendars were exposed in their most important buildings like the World-wide famous Temple of Kukulkan (“Feathered serpent”) in the archeological site of Chichen Itza. The temple was founded around 525 AD although the current building was completed between the 9th and the 12th centuries. The pyramid has four sides, each one with 91 steps, which adds up to 364 steps. If we count the last platform as a step we get 365 steps, which is equal to the days we find in the Haab calendar.

Photography by Roberto Lara

But the most famous thing about the Kukulkan’s temple is the descent of Kukulkan: during the autumn and spring equinoxes the late afternoon Sun strikes off the northwest corner of the pyramid and casts a series of triangular shadows against the northwest balustrade, creating the illusion of a feathered serpent ‘crawling’ down the pyramid. We should remark that the balustrade and corners of the pyramid are perfectly aligned, which makes us admire even more the work that Mayans had on the building:

The Feathered Serpent in the Spring Equinox
Source: Wikimedia Commons

The pyramid also shows us that Mayans had some knowledge about acoustics. If you stand in front of any of the four stairway and clap your hands, the pyramid reflects the sound in such way that you hear the sing of a quetzal, a bird from the jungle. It’s fascinating! Isn’t it? Moreover, the shaman was known as ‘the man with the great voice’, because when people met for a ritual, he didn’t have to speak loudly, as everybody could hear him perfectly.

From all these facts, we can easily conclude that mathematics in the ancient Mayan world wasn’t only a help for agriculture but a tool through which the leaders could control the population. In fact, in the picture below we can see the ruins of a Mayan school. Only those from the upper class had access to the education, and we can see from the building they truly wanted to keep it as a privilege!

Photography by Roberto Lara

The hole at the right of the picture was made by an adventurer who thought gold was hiding inside it and used dynamite to enter the building.

This post has been written by Roberto Lara Martín in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).

Location: Chichen Itzá (map)

# Al-Bîrûnî’s birthday

Abû al-Rayhân Muhammad ibn Ahmad al-Bîrûnî was born near Kath in the region of Khwârazm (now Kara-Kalpakskaya) in September 4, 973, and was died in Gbazna(?) after 1050. He lived in Kath and in Jurjanîyya and we know that he began his studies under Abû Nasr Mansûr (970-1036). He became a good mathematician and astronomer very fast and he measured the latitude of Kath observing the maximum altitude of the Sun when he was only 17 y.o. He also wrote some astronomical and mathematical works before 995 as we can check in his  Cartography (a book about map projections).

About 995, al-Bîrûnî left the civil war in Khwârazm and moved to Rayy (now near Tehran) where he lived in poverty. We know that he worked with al-Khujandî who had a large sextant with which he had determined the obliquity of the ecliptic.

Through the observational data registered by him we know that he spent some days in Rayy and that he was back in his birthplace in 1004. That year he became protected by the rulers of the region and he got enough money to build an instrument at Jurjanîyya to observe solar meridian transits. He made observations with it in 1016 and one year later he and Abû Nasr Mansûr were made prisoners by Mahmûd, the new ruler of Khwârazm. Al-Bîrûnî continued working as astronomer but he had a lot of problems with victorious Mahmûd. However, between 1018 and 1020 he made observations from Ghazna supported by Mahmûd and this work allowed him to determinate the latitude of the place.

Al-Bîrûnî travelled to India together with Mahmûd’s military expedition and he spent at least five years working on his India, in which he computed latitudes of cities and explained calendars, geography, literature…

After Mahmud’s death, the next rulers allow al-Bîrûnî to be free to travel and work in his interests and he became the most prolific Arabic mathematician in the World. He produced 146 works with more than 13.000 pages!

Russian stamp (1973)

So, al-Bîrûnî’s doodle published two years ago in the Arabic countries is a very good example of a great scientific contribution!

# Zu Chongzhi’s birthday

Zu Chongzhi (429-501) is one of the greatest ancient Chinese mathematicians. He was taught Mathematics from Liu Hui’s commentary on the Nine Chapters on the Mathematical Art. He was appointed as an officer in the city of Yang-chou and during this time he worked in Arithmetic and Geometry. He gave the rational 355/113 as an approximation of pi in his Zhui shu (“Method of interpolation”) and proved that:

3,1415926 < pi < 3,1415927

Zu also proposed a new calendar (the “Calendar of Great Brightness”) in 462 based on a cycle of 391 years with 144 extra months inserted (=4.836 months).

In the Wikimedia Commons there is this photo of a statue dedicated to him in Tinglin Park in Kunshan.

Source: Wikimedia Commons

Google dedicated this doodle five years ago to celebrate his birthday.

Location: Tinglin Park in Kunshan (map)

# The Royal Palace of Madrid

Source: Wikimedia Commons

The Royal Palace of Madrid isn’t a mathematical place. Today I have visited it but I haven’t found mathematics in the interesting tour through its rooms. The palace was built by Philip V of Spain on the site of a former palace which was partially destroyed by the fire in 1734. The first Spanish king who lived in the palace was Charles III (Philip V’s third son) and Charles IV, Joseph Bonaparte, Ferdinand VII, Elisabeth II, Alfonso XII and Alfonso XIII also lived here.

I must recognize that I couldn’t leave the Royal Palace without getting a picture to write this post and finally I’ve got two “astronomical” details that justify these sentences. The first one is located on the facade of the palace and it’s an explicit reference to the horoscope:

Detail of the facade of the Royal Palace
Photography by Carlos Dorce

Detail of the facade of the Royal Palace
Photography by Carlos Dorce

The other interesting object is a clock which can be found in the small Porcelain Room next to the room where the king Charles III died in 1788: there is a wonderful clock representing Atlas holding the World:

Photography by Carlos Dorce

And that’s all! Tomorrow I’m comig back to Barcelona but I’m sure that I’m going to talk about Madrid for some days.

Location: Royal Palace of Madrid (map)

# End of the Mayan Calendar

Today is the end of the World!

Sorry, it is only the end in the Mayan calendar and Google has dedicated its doodle today!

# The House of the Blackheads

Photography by Carlos Dorce

The House of the Blackheads (Melngalvju nams) is located in the heart of the old town of Riga. It’s a building erected in the first half of the 14th century for the Brotherhood of Blackheads who was an association of local unmarried ship owners and merchants founded in the 14th century. The house is one of the most folkloric buildings in the city because of its ornamentation built during the XVIth, XVIIth and XVIIIth centuries.  In the Second World War, the house was destroyed by the German bombs (1941) and the current picture was built in the period 1995-1999.

House of the Blackheads in 1891
Source: Wikimedia Commons

I’ve been in Riga twice and I’ve always admired this beautiful house! And the astronomical clock on its front!

Astronomical clock
Photography by Carlos Dorce

The clock has three circles. The upper one indicates whether it is day or night and the the lower one points to the day of the week. The main clock is a calendar (days and months) and a analogical clock (hours and minutes). I want to return to Prague to enjoy its great astronomical clock but I think that this one can be just as interesting as the Czeck one!

Location: The House of the Blackheads (map)

# The Tro-Cortesianus Codex

The Tro-Cortesianus Codex

In 1880, Léon de Rosny figured out that the two extant parts of the Tro-Cortesianus Codex (also know as the Madrid Codex) were a single codex and Spaniards were very lucky to have one of the most important codices of the Maya civilization because it’s the longest of the surviving Maya codices. The Museo Arqueologico Nacional acquired it from book-collector José Ignacio Miró in 1872 who had purchased it in the Spanish region called Extremadura a few years before. The director of the Museum decided to name the Cortesianus Codex after Hernán Cortés (Medellín, 1485 – Castilleja de la Cuesta, 1547) supposing that he had brought the codex to Extremadura.

Portrait of Hernán Cortés
Galleria degli Uffizi (Florence)

The Codex is dated in the period 1250-1500 and it consists in 56 sheets painted on both sides to produce a total of 112 pages. Each page measures roughly 23.2 by 12.2 centimetres (9.1 by 4.8 in). The content consists of horoscopes, astronomical tables and almanacs used by the priests in the performance  of their ceremonies and rituals. The Mayans reached a precise idea of the movements of the Sun, the Moon and the planets and they estimated at 584 days the synodic revolution of Venus (the cycle is really equal to 583,92 days). They also realized that the solar year consisted of approximately 365,242 days. So the synchronization error between the Venus cycle, the solar year and the liturgical year was only one day every 6,000 years!

Mayan priest observing the night sky
Tro-Cortesianus Codex

We can also see a lot Mayan numbers so we must notice how Mayans wrote their numbers. Mayan numbers consist in series of points and horizontal (or vertical) lines. Each line represent 5 and each point represents 1. For example, in the next page:

Page 93
Tro-Cortesianus Codex

The page is divided in four images and in the lower one we can see numbers 3, 7, 11, 8, 6, 8, 13 and 1. Now you can explore the other parts and say which numbers are in each one!

Finally, we can talk about the Mayan calendar from two pages of the codex and the representation of some Mayan numbers and the famous glyphs. We can see in the middle of the picture a torch inside a square decorated with the 20 glyphs corresponding to the fundamental set of 20 successive days: Imix, Kimi, Chuwen, Kib, Ik, Manik, Eb, Kaban, Akbal, Lamat, Ben, Etsnab, Kan, Muluc, Ix, Kawak, Chikchan, Ok, Men and Ahaw.

Pages 95-96

Each day was represented by a different glyph and they were related to gods and sacred animals and objects: Imix was related to the crocodile and the nenuphar, Kimi to the Death’s divinity,… :

Mayan glyphs

In the liturgical calendar, the 20 days used to be also related to the 13 first natural numbers in a cyclical period. Firstly, the first day was related to 1, the second to 2, the third to 3,… and the thirteenth to 13. Then, the fourteenth was related to 1 again, the fifteenth to 2,… and the twentieth to 7. The first day of the second cycle was related to 8 and so on. After 13 complete cycles (= 13 · 20 days = 260 days) the first day of the cycle was related to 1 again.

Outside the square, the upper section of the picture represents the East (Lakin), the right part is the North (Xaman), the left part is the South (Nokhol) and the lower part is the West (Chikin). We can see perfectly five pairs of glyphs representing the days in each of the four corners and there also are 20 sets of 13 points representing the different cycles of the Mayan calendar. All the glyphs have the number 1 or the number 13 with them. So we are reading a Mayan liturgical calendar through the 260 points representing the 260 days!

Location: Museo de América at Madrid (map)

# A Runic calendar in Tallinn

Estonian Runic Calendar (1819)
Estonian History Museum (Tallinn)
Photography by Carlos Dorce

A Runic calendar is a perpetual calendar based in the 19 year Metonic cycle of the Moon. The Greek astronomer Meton of Athens (Vth century BC) observed that a period of 19 years was equal to 235 synodic months and 6.940 days which is almost equal to 19 solar years except for a few hours. This cycle was used in the Babylonian calendar and Meton computed all the necessary parameters and the intercallary months to adjust the periods of the Sun and the Moon.

Runic calendars were written on parchment or carved onto staves of wood (as the one of the Estonian History Museum), horn or bone. It appears to be  a medieval Swedish invention and the Nyköping staff, believed to date from the 13th century, is the oldest one which is preserved. The Runic calendar preserved in the Museum is dated in 1819 and its first line is made up of the first seven letters of the Runic alphabet (runes). 52 weeks of 7 days were laid out using 52 repetitions of this first seven runes and each rune corresponded to each weekday varied from year to year. On another line, many of the days were marked with one of the 19 symbols representing the 19 possible positions of a year in the Metonic cycle (called “Golden Numbers”).

This kind of calendars were used until mid-19th century.