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Mathematical Tourism in Siberia George Lugosi

Does your hometown have any

mathematical tourist attractions such

as statues, plaques, graves, the cafd

where the famous conjecture was made,

the desk where the famous initials

are scratched, birthplaces, houses, or

memorials? Have you encountered

a mathematical sight on your travels?

If so, we invite you to submit to this

column a picture, a description of its

mathematical significance, and either

a map or directions so that others

may follow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: dirk.huylebrouck@ping.be

I n Western Siberia, in the lower basins of the rivers Ob and Irtysh, live the

Khants (or Hants, Hanti, or Ostyaks). Although they are but a small group of about 20,000 to 23,000 people, they are the third largest group of the North. The region is not very densely popu- lated, as the territory is unforgiving ter- rain, inundated by snow and ice in win- tertime and by water in summertime.

In this harsh local climate, the Khants' log-houses, covered by birch- bark and raised on stilts, provide suffi- cient shelter. Clay ovens surround them, and for centuries, bread was baked and fish smoked in the same tra- ditional way. However, since the 1960s, land devastation by ruthless oil or in- frastructure development and the con- sequent pollution has caused an eco- nomic and social crisis. It severely threatens the Khant culture. It is timely to study what is left of their culture, and in particular the aspects of inter- est to mathematicians.

Their folk art is remarkable. Pat- terns on coats and dresses are of particular interest (Fig. 3). The rein- deer-skin fur is t r immed in part with reindeer and in part with hare fur, cut to an even length. The names of the decorative band patterns are "swan's legs" and "hare's ears." In several areas (especially in Eastern and South Eu- rope), coat decorat ion determines whether the garment belongs to a girl or to a woman, but Khant folk art does not make such a distinction, and the topcoats of girls of marriageable age and married women hardly differ.

We can describe certain patterns in shirts like the one in Figure 3 in terms of permutation. Indeed, the cross- stitch patterns on the shirts can be summarized in a matrix

M = (m i j ) i j 1 . . . . . n

with values 0 (white) or 1 (black). To the left of each row, a number indicates how many ones there are in the corre- sponding row, and above each column,

another number specifies the number of ones in the corresponding column. In these patterns the mij element will be 1 when the sum of the numbers to the left of the row i and above the col- umn j is greater than n.

For instance, if n = 4, and if 1, 2, 3, 4 designates the desired number of ones in the respective rows, and 4, 3, 2, 1 the desired number in the columns, then ml,z = 1 + 4 (>n) , ml,2 = 1 + 3 (=n), ml,3 = 1 + 2 (-<n) . . . . This yields the configuration

4 3 2 1 1 2

3

4

In the same way, the indication 3, 2, 1, 4 left of the rows and 4, 3, 1, 2 above the columns, produces

4 3 1 2

3 2 1 4

To generate all of the configurations possible using the same column se- quence, a diagram is made as follows. Starting with the row sequence 1, 2, 3, 4, the permutat ion of rows provides 24 = 4? possibilities of Figure 4. Per- muting the column sequence, another 24 possibilities result in 24 tables, each containing 24 configurations. A block with all possibilities will thus contain 576 = 4? • 4? different configurations. In every table, some configurations are

3 4 THE MATHEMATICAL INTELLIGENCER �9 2004 SPRINGER VERLAG NEW YORK, LLC

Figure 1, Map showing the Khant region.

Figure 2. A. Y. Filtchenko's photograph of a traditional Khant log-house (1998).

Figure 3. A shirt with rich cross-stitched patterns. The arrows indi-

cate patterns explained in the text (Photo: Markku Haverinen).

VOLUME 26, NUMBER 2, 2004 3 5

Figure 4.

September 29, 2003, the museum or- ganized a special exposition titled: "Sibe- ria. Life on the Taiga and Tundra" (see http://www.nba.fi/NEWS/LEHDISTO/ 2002/siperialehdistokuvat.htm). The mu- seum is housed in a former indoor ten- nis complex, at the "Etel~iinen Rauta- tiekatu 8 / Salomonkatu 15" in Helsinki. The multiplex has been renovated to house the museum as well as the Helsinki City Art Museum, Finnkino's cinema centre, and a number of shops, restaurants, and caf6s.

R&D&l--Research, Develop, Invent

2 Union Street

Kew 3101

Victoria, Australia

e-mail: g.lugosi@hfi.unimelb.edu.au

Figure 6.

Figure 5.

the horizontal mirror images of each other, whereas in each block the con- figurations of some tables are vertical mirror images. The pattern of the con- nections, valid inside the tables and in- side the blocks, is shown in Figure 5.

Figure 6 is a magnified image of Fig- ure 3; it uses repeatedly the configura- tion 1, 3, 2, 4 (rows) and 3, 1, 4, 2 (columns) for n = 4, and the configu- ration 1, 4, 2, 5, 3 (rows) and 5, 1, 3, 2, 4 (columns) for n = 5, along with their mirror images.

To investigate the occurrence of

these patterns in different regions of Siberia (see Figure 7 for a more com- plex example), or elsewhere in the world, is an interesting ethno-mathe- matical subject. The mathematical set- up here may help to classify different stitched patterns and to compare con- figurations made by cultures from dif- ferent regions or from different eras.

A museum that attaches great im- portance to these fur coats with their intriguing geometrical patterns is the Helsinki Museum of Cultures. Re- cently, between May 14, 2002, and

Figure 7. More cross-stitched Khant pat-

terns.

36 THE MATHEMATICAL INTELLIGENCER