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A Fake Circle made of Straight Lines
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
Polygon approximation era
can be estimated by computing the perimeters of circumscribed and inscribed polygons.
The first recorded algorithm for rigorously calculating the value of Ï was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes. This polygonal algorithm dominated for over 1,000 years, and as a result Ï is sometimes referred to as "Archimedes' constant". Archimedes computed upper and lower bounds of Ï 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 < Ï < 22/7 (that is 3.1408 < Ï < 3.1429). Archimedes' upper bound of 22/7 may have led to a widespread popular belief that Ï is equal to 22/7. Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for Ï of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of Ï in 1630, a record only broken in 1699 when infinite series were used to reach 71 digits.
developed the polygonal approach to approximating Ï
In ancient China, values for Ï included 3.1547 (around 1 AD), â10 (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 Ï of 3.1416. Liu later invented a faster method of calculating Ï 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 Ï â 355/113 (a fraction that goes by the name MilÃ¼ 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 Ï available for the next 800 years.
The Indian astronomer Aryabhata used a value of 3.1416 in his Äryabhaá¹Ä«ya (499 AD).Fibonacci in c. 1220 computed 3.1418 using a polygonal method, independent of Archimedes. Italian author Dante apparently employed the value 3+â2/10 â 3.14142.
The Persian astronomer JamshÄ«d al-KÄshÄ« produced 9 sexagesimal digits, roughly the equivalent of 16 decimal digits, in 1424 using a polygon with 3Ã228 sides, which stood as the world record for about 180 years. French mathematician FranÃ§ois ViÃ¨te in 1579 achieved 9 digits with a polygon of 3Ã217 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, Ï 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 1040 sides, which remains the most accurate approximation manually achieved using polygonal algorithms.
- ^ Arndt & Haenel 2006, p. 170
- ^ Arndt & Haenel 2006, pp. 175, 205
- ^ "The Computation of Pi by Archimedes: The Computation of Pi by Archimedes â File Exchange â MATLAB Central". Mathworks.com. Archived from the original on 25 February 2013. Retrieved 12 March 2013.
- ^ Arndt & Haenel 2006, p. 171
- ^ Arndt & Haenel 2006, p. 176
Boyer & Merzbach 1991, p. 168
- ^ Arndt & Haenel 2006, pp. 15â16, 175, 184â186, 205. Grienberger achieved 39 digits in 1630; Sharp 71 digits in 1699.
- ^ Arndt & Haenel 2006, pp. 176â177
- ^ a b Boyer & Merzbach 1991, p. 202
- ^ Arndt & Haenel 2006, p. 177
- ^ Arndt & Haenel 2006, p. 178
- ^ Arndt & Haenel 2006, pp. 179
- ^ a b Arndt & Haenel 2006, pp. 180
- ^ Azarian, Mohammad K. (2010). "al-RisÄla al-muhÄ«tÄ«yya: A Summary". Missouri Journal of Mathematical Sciences. 22 (2): 64â85. Archived from the original on 14 January 2015.
- ^ 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.
- ^ a b c Arndt & Haenel 2006, p. 182
- ^ Arndt & Haenel 2006, pp. 182â183
- ^ a b Arndt & Haenel 2006, p. 183
- ^ 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 < Ï < 3.14159 26535 89793 23846 26433 83279 50288 4199.