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Objects of Mathematical Interest at the MFA    

This page discusses some objects on display at the Museum of Fine Arts in Boston, that have some interesting mathematical property.

Vessel in the shape of a dragon-tortoise
Vessel in the shape of a dragon-tortoise

Vessel in the shape of a dragon-tortoise


Joseon Dynasty, 18th century


Porcelain with molded decoration with details painted in underglaze blue

18.6 x 24 x 26.5 cm (7 5/16 x 9 7/16 x 10 7/16 in. )

Museum of Fine Arts, Boston: Charles Bain Hoyt Collection 50.2212

Provenance: By 1948, Charles Bain Hoyt (b. 1889 - d. 1949), Camden; 1950, bequest of Hoyt to the MFA (Accession Date: May 11, 1950).

1st floor, Asian wing, Korean gallery

On the turtle's back are 9 groups of dots, each with a different number from one to 9. The dots represent numbers; if the turtle's head is at the top, the numbers are arranged like so:

4 9 2
3 5 7
8 1 6

(In the above picture, only five of the 9 numbers can be seen).

This pattern of numbers is a 3x3 magic square. All the rows, columns, and diagonals ass up to 15. It was well-known in antiquity and is seen in many cultures. The specific connection with a turtle (or tortoise) comes from a story in Lo Shu, the book of the river Lo, which dates from China roughly 1000 BCE.

There was a huge flood. The people tried to offer some sacrifice to the 'river god' of one of the flooding rivers, the 'Lo' river, to calm his anger. However, every time a turtle came from the river and walked around the sacrifice. The river god didn't accept the sacrifice until one time, a child noticed the curious figure on the turtle shell. Hence they realized the correct amount of sacrifice to make.

A 4x4 magic square appears in the Dürer woodcut Melancholia, shown elsewhere on this page.

Melencolia I
Melencolia I

Albrecht Dürer, German, 1471-1528

Melencolia I



Platemark: 23.9 x 16.8 cm (9 7/16 x 6 5/8 in.)

Bartsch (Intaglio) 074; Meder 75, II, c(?)

Museum of Fine Arts, Boston: Centennial Gift of Landon T. Clay 68.188

Provenance: P. Lely (Lugt 2092); Tomás Harris (England, 1908-1964); purchased through P. & D. Colnaghi & Co., Ltd., London by MFA April 10,1968

currently not on view

This engraving is a sort of still life collecting a large number of images and symbols related to the "science" of alchemy, the precursor to modern chemistry.

In the upper-right corner of this print is a 4x4 magic square. Here is a detail:

The 4x4 square (detail)
The 4x4 square (detail)

A few of the numerals are different from the modern form; the numbers are:

16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1

Every row, column and diagonal adds up to 34; also the four corners add to 34, the four in the center add to 34, and the four in any one quadrant (e.g. 16+3+5+10) also add to 34.

Also notice the two numbers 15 and 14 in the bottom row, which together form 1514, the date the work was created.

A 3x3 magic square appears on the back of the Korean tortoise vessel shown elsewhere on this page.

You can also see a perspective illusion in the ladder: the bottom of the ladder appears to be closer than the structure that the ladder is leaning against, but the top of the ladder is clearly behind that structure. Such illusions are best known from the works of M. C. Escher.


Star and Cross Tiles

There are several of these displayed together near the west end of the Islamic corridor on the first floor. Tiles of these two shapes (combined) can be used to tessellate an arbitrarily large 2-dimensional space.

Under the Wave off Kanagawa (Kanagawa-oki nami-ura), from the series Thirty-six Views of Mount Fuji (Fugaku sanjurokkei)

Katsushika Hokusai


Edo period, about 1830-31


Woodblock print (nishiki-e); ink and color on paper

Horizontal oban; 25.8 x 38 cm

Museum of Fine Arts, Boston: William S. and John T. Spaulding Collection 21.6765

Provenance: Spring 1913, purchased by William S. and John T. Spaulding from Frank Lloyd Wright in Japan; December 1, 1921, given by William S. and John T. Spaulding to the Museum.

Also known as "The Great Wave Off Kanagawa", this famous woodblock is often given as an illustration of the self-similar quality of fractals, a special class of geometric objects involving a dimension between 1 and 2 (or between 2 and 3). The curly point of the wave is made of several smaller curly points, each of which has several smaller curly points and so on. The appearance of the wave is similar to that of the "Heighway" dragon curve fractal.

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