Beijing is a city of 10 000 000 people and 5 000 000 bicycles. Formerly known as Peking or Peiping, it lies 183 km inland on the edge of the North China plain. It is prone to coal dust pollution and to dust storms early in the year; many people may be seen wearing surgical masks as protection. Since it is hot (38°C) in summer and very cold in winter (-20°C), it is best to see Beijing in autumn. Many foreigners and Chinese visit Beijing to see the Great Wall of China, and the Forbidden City: a 100 hectare moated, walled city-within-a-city built as the palatial residence of the 15th century Ming emperors. Beijing also offers the visitor the best museums and the most impressive palaces, temples and monumental buildings in China.

One can travel by bus, subway, train or, if adventurous and fit, by bicycle. It is, however, crowded, whichever way one wishes to travel!
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The Temple of Heaven
(Tian Tan)

Among the sights not to be missed, and within walking distance of the Forbidden City and most major hotels, is the Temple of Heaven (Tian Tan), a masterpiece of 15th century Chinese architecture.

The temple was designed according to the most advanced principles of mechanics, geometry and fengshui (geomancy), and incorporates a wealth of examples of number, symmetry and geometric shape.

It was the site of the most sacred ceremonies conducted by the Emperor (the Son of Heaven) during the winter solstice and at other select times of the year. It is situated within a large park, and consists of three main parts.































Parts of the Temple

Hall of Prayer for Good Harvests (Qiniandian). This, the main temple, built in 1420 A.D., consists of cones and cylinders. The four main pillars, the 12 interior posts and the 12 pillars in the outer wall represent the four seasons, the 12 months of the year and the 12 ‘hours’ of the Chinese day. The elaborate decoration contains many examples of numerical significance, and much symmetry.

Imperial Vault of Heaven. This is a simpler version of the Hall of Prayer, built in 1530 A.D. It has a circular wall around the outer courtyard with the curious acoustic property that people standing at diametrically opposite points can hear each other speak in a whisper. In the paving of the central stairway, there are three large stones known as the Three Echo Stones. A sharp noise such as a hand clap produces a single echo from the first stone, a double echo from the second stone and a triple echo from the third.

Circular Mound Altar of Heaven. Built in 1539 A.D., this consists of three terraces in the form of concentric discs atop each other representing the earth, the mortal world and Heaven. Flagstones on the upper terrace are arranged in concentric circles in multiples of nine (nine is the number signifying Heaven); the innermost circle has nine stones, the second 18, and so on to the 27th circle.

 Investigate How many flagstones has the 27th circle?






















The Chinese calendar

The Hall of Prayer contains much symbolism reflecting Chinese interest in the calendar. The oldest Chinese calendar (2357 B.C.) marks the bstart of the first 60-day cycle used to reckon the days, moons and years in Chinese chronology. The name of each day of the 60-day cycle was formed from the names of the characters in a subcycle of 10 celestial signs:

1. Kia  2. I  3. Ping  4. Ting  5. Wu  6. Ki  7. Kêng  8. Hsin  9. Jên  10. Kuei,

together with the names of characters in a subcycle of 12 terrestrial signs:

1. Tzu  2. Chou  3. Yin  4. Mao  5. Shin  6. Ssu  7. Wu  8. We  9. Shin  10. Yu  11. Hsü  12. Hai.

The first day of the 60-day cycle was named by combining the name of the first celestial sign with that of the first terrestrial sign (thus Kia-Tzu). The second day was I-Chou, the third, Ping-Yin, the 11th Kia-Hsü, the 12th I-Hai, and the 13th Ping-Tzu.

   Investigate Use a matrix array to name of the 60 th (last) day of the cycle.

This Chinese calendar was ‘lunisolar’: months were determined by the moon, and length of year by the sun. As there are about 29.5 days between successive full moons, the Chinese year was divided into 12 months of 29 or 30 days to produce a (lunar) year of 354 days.

 Investigate Use your library or the Internet to find out more about the Chinese calendar with its 12 year cycle.

 






















Chinese frieze patterns

Frieze patterns feature in the Hall of Prayer, and are part of Chinese culture. On Chinese walls, fabrics, pottery etc, there are a number of distinct motifs forming frieze patterns. Below are some examples. Each has its own symmetry generated by a translation, sometimes combined with a reflection in a vertical or horizontal axis, or a half-turn. (A translation followed by a reflection in a line of translation is a glide reflection).

 Investigate In each pattern, find a motif, a smallest portion of the pattern, from which the whole pattern can be generated. A motif need not be unique.

Investigate the symmetries of these patterns by locating axes and centres of symmetry both within and between motifs.

 
The above patterns have names. From the top: pearl border,
T pattern and key pattern,
swastika pattern, cloud border, still water, and sea waves.




























The seven frieze patterns

There are essentially only seven distinct frieze patterns which are classified according to the symmetries which generate them. They are illustrated in the table below. A ‘horizontal’ reflection is reflection in a horizontal mirror line.
Motif   Pattern
Symmetries

Horizontal translations, (through the width of an L).

Horizontal translations and vertical reflection.

Horizontal translations, half-turn about centre of an H, horizontal reflection, and two distinct vertical reflections.

Horizontal translations, and two distinct vertical reflections.

Horizontal translations and horizontal glide reflections.

Horizontal translations, two distinct vertical reflections, and horizontal glide reflections.

Horizontal translations, and half-turns within and between motifs.

 Project   Find and classify some modern day friezes. You may also wish to construct some designs of your own.