A Fake Circle made of Straight Lines

By Monty, 15th June 2015

Inspired by this post on Reddit, I wrote this little script to do the same thing. The circle is an illusion created by 100 straight-line tangents to an invisible circle. The tangents are created by drawing a chord in a larger concentric circle, and moving it’s endpoints around by equal increments 100 times. The larger circle’s circumference is defined by x=sin(t), y=cos(t), for t in the range 0 to 2*pi. So a chord’s endpoints will be defined by 2 different values of t.

  • Import turtle & math libraries
  • Set r (radius) to 400 px.
  • Set angular increment to a 100th of the circle, i.e. 2*pi/100
  • Initialise the plotting angle to to 0, and an offset to something in the range 0:2*pi
  • Make 100 steps around the circle
    • compute endpoints (x1,y1), (x2,y2) of a chord
    • lift pen and goto first endpoint; drop it & goto second
    • increment angle
Wikipedia:

Polygon approximation era

diagram of a hexagon and pentagon circumscribed outside a circle
π can be estimated by computing the perimeters of circumscribed and inscribed polygons.

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.[1] This polygonal algorithm dominated for over 1,000 years, and as a result π is sometimes referred to as "Archimedes' constant".[2] 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).[3] Archimedes' upper bound of 22/7 may have led to a widespread popular belief that π is equal to 22/7.[4] 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.[5] 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.[6]

A painting of a man studying
Archimedes developed the polygonal approach to approximating π.

In ancient China, values for π included 3.1547 (around 1 AD), 10 (100 AD, approximately 3.1623), and 142/45 (3rd century, approximately 3.1556).[7] 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.[8][9] 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.[8] The Chinese mathematician Zu Chongzhi, around 480 AD, calculated that π355/113 (a fraction that goes by the name Milü in Chinese), using Liu Hui's algorithm applied to a 12,288-sided polygon. With a correct value for its seven first decimal digits, this value of 3.141592920... remained the most accurate approximation of π available for the next 800 years.[10]

The Indian astronomer Aryabhata used a value of 3.1416 in his Āryabhaṭīya (499 AD).[11] Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes.[12] Italian author Dante apparently employed the value 3+2/10 ≈ 3.14142.[12]

The Persian astronomer Jamshīd al-Kāshī produced 9 sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424 using a polygon with 3×228 sides,[13][14] which stood as the world record for about 180 years.[15] French mathematician François Viète in 1579 achieved 9 digits with a polygon of 3×217 sides.[15] Flemish mathematician Adriaan van Roomen arrived at 15 decimal places in 1593.[15] 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).[16] Dutch scientist Willebrord Snellius reached 34 digits in 1621,[17] and Austrian astronomer Christoph Grienberger arrived at 38 digits in 1630 using 1040 sides,[18] which remains the most accurate approximation manually achieved using polygonal algorithms.[17]

  1. ^ Arndt & Haenel 2006, p. 170
  2. ^ Arndt & Haenel 2006, pp. 175, 205
  3. ^ "The Computation of Pi by Archimedes: The Computation of Pi by Archimedes – File Exchange – MATLAB Central". Mathworks.com. Archived from the original on 25 February 2013. Retrieved 12 March 2013. 
  4. ^ Arndt & Haenel 2006, p. 171
  5. ^ Arndt & Haenel 2006, p. 176
    Boyer & Merzbach 1991, p. 168
  6. ^ Arndt & Haenel 2006, pp. 15–16, 175, 184–186, 205. Grienberger achieved 39 digits in 1630; Sharp 71 digits in 1699.
  7. ^ Arndt & Haenel 2006, pp. 176–177
  8. ^ a b Boyer & Merzbach 1991, p. 202
  9. ^ Arndt & Haenel 2006, p. 177
  10. ^ Arndt & Haenel 2006, p. 178
  11. ^ Arndt & Haenel 2006, pp. 179
  12. ^ a b Arndt & Haenel 2006, pp. 180
  13. ^ Azarian, Mohammad K. (2010). "al-Risāla al-muhītīyya: A Summary". Missouri Journal of Mathematical Sciences. 22 (2): 64–85. Archived from the original on 14 January 2015. 
  14. ^ O'Connor, John J.; Robertson, Edmund F. (1999). "Ghiyath al-Din Jamshid Mas'ud al-Kashi". MacTutor History of Mathematics archive. Archived from the original on 12 April 2011. Retrieved 11 August 2012. 
  15. ^ a b c Arndt & Haenel 2006, p. 182
  16. ^ Arndt & Haenel 2006, pp. 182–183
  17. ^ a b Arndt & Haenel 2006, p. 183
  18. ^ Grienbergerus, Christophorus (1630). Elementa Trigonometrica (PDF) (in Latin). Archived from the original (PDF) on 2014-02-01.  His evaluation was 3.14159 26535 89793 23846 26433 83279 50288 4196 < π < 3.14159 26535 89793 23846 26433 83279 50288 4199.
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