section_path: History/Polygon approximation era
section_name: Polygon approximation era
paragraph_text: Polygon approximation era
  
  The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes, implementing the method of exhaustion. This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as Archimedes's constant. Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By calculating the perimeters of these polygons, he proved that 223 / 71 < π < 22 / 7 (that is, 3.1408 < π < 3.1429). Archimedes' upper bound of 22 / 7 may have led to a widespread popular belief that π is equal to 22 / 7. Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of π in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.
  
  In ancient China, values for π included 3.1547 (around 1 AD), 10 {\displaystyle {\sqrt {10}}} (100 AD, approximately 3.1623), and 142 / 45 (3rd century, approximately 3.1556). Around 265 AD, the Wei Kingdom mathematician Liu Hui created a polygon-based iterative algorithm and used it with a 3,072-sided polygon to obtain a value of π of 3.1416. Liu later invented a faster method of calculating π and obtained a value of 3.14 with a 96-sided polygon, by taking advantage of the fact that the differences in area of successive polygons form a geometric series with a factor of 4. The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that 3.1415926 < π < 3.1415927 {\displaystyle 3.1415926<\pi <3.1415927} and suggested the approximations π ≈ 355 113 = 3.14159292035... {\textstyle \pi \approx {\frac {355}{113}}=3.14159292035...} and π ≈ 22 7 = 3.142857142857... {\textstyle \pi \approx {\frac {22}{7}}=3.142857142857...}, which he termed the Milü ('' close ratio") and Yuelü ("approximate ratio"), respectively, using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value remained the most accurate approximation of π available for the next 800 years.
  
  The Indian astronomer Aryabhata used a value of 3.1416 in his Āryabhaṭīya (499 AD). Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes. Italian author Dante apparently employed the value 3 + 2 10 ≈ 3.14142 {\textstyle 3+{\frac {\sqrt {2}}{10}}\approx 3.14142}.
  
  The Persian astronomer Jamshīd al-Kāshī produced nine sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424, using a polygon with 3 × 2 28 {\textstyle 3\times 2^{28}} sides, which stood as the world record for about 180 years. French mathematician François Viète in 1579 achieved nine digits with a polygon of 3 × 2 17 {\textstyle 3\times 2^{17}} sides. Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593. In 1596, Dutch mathematician Ludolph van Ceulen reached 20 digits, a record he later increased to 35 digits (as a result, π was called the "Ludolphian number" in Germany until the early 20th century). Dutch scientist Willebrord Snellius reached 34 digits in 1621, and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630 using 10 sides. Christiaan Huygens was able to arrive at 10 decimal places in 1654 using a slightly different method equivalent to Richardson extrapolation.