As you can see, the Chapel of the Shroud of Turin is between the west wing of the Royal Palace and the back of the Cathedral, and is elevated above its high altar.
Unfortunately the chapel can't be visited right now: it has been under restoration for more than 20 years, since the night when a fire that started in the attics of the Royal Palace heavily damaged it. The whole building was blackened by smoke.
The Shroud was saved thanks to the firefighters' intervention, who promptly brought it to salvation by breaking through the glass display case it was kept in.
The chapel is one of the most extravagant masterpieces of Baroque architecture in Italy. It was commissioned by Charles Emmanuel of Savoy, and originally called for a normal circular shape surmounted by a dome. But in 1668 the genius architect Guarino Guarini decisively revolutionized the existing structure, introducing six large windows alternating with projections in the wall. A series of arches connects the openings to the projections, and the top of the arches is connected to the center of the dome, forming a hexagon. The interlacing arches creates a kind of nest that reaches the top of the dome, where the dove of the Holy Spirit emerges at the center of a twelve-pointed star. The great central altar with the silver and crystal shrine was built at the end of the 18th century to house the Shroud. The colored marble floor with geometric drawings and bronze stars is also worth mentioning. The walls display funerary monuments of some Savoy family members.
If you look at it from the outside, the dome makes you think of the architecture from the Far East: the curved lines create a variety of semicircle openings, giving the impression of an embracing, almost spiral movement.
FUN FACT: in addition to being an architect, Guarino Guarini was also a great mathematician. The chapel of the Shroud is a play on the number 3 and its multiples.
Outside there are 6 windows, 12 arches, and 3 steps under the lantern. Inside its 3 rooms, there are columns in groups of 3 and 3 niches. The 3 of course refers to the Trinity, but also to the number of days of the Passion.