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Alexander Zouev
000051 - 060 - 3 -
Extended Essay – Mathematics
Alhazen’s Billiard Problem
Introduction:
Regarded as one of the classic problems from two dimensional geometry,
Alhazen’s Billiard Problem has a truly rich history. The problem is believed to have been first introduced by Greek astronomer Ptolemy back in 150 AD1 and then eventually noticed by 17th century Arabic mathematician Abu Ali al Hassan ibn Alhaitham (whose name was later Latinized into Alhazen)2
.
Alhazen made reference to this problem in one of his published works entitled
Optics and presented it in the form, “Find the given point on a spherical mirror at which a ray of light coming from a given point must strike in order to be reflected” 3. Nowadays, this problem is often referred to as the “Billiard Problem” because it involves locating the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in order to carom (bounce) once off the edge and strike another ball at a second given point.4 The focus question of this extended essay will be: Heinrich Dörrie also described the problem as “find in a given circle an isosceles On a circular billiards table there are two balls; at what point along the circumference must one be aimed at in order for it to strike the other after rebounding off the edge?
1
Jack Klaff, “The World May be Divided into Two Types of People – Alhazen’s Billiard Problem.”
Viewed 19 February 2005.
2
Heinrich Dörrie, 100 Great Problems of Elementary Mathematics: Their History and Solutions. Dover
Publications New York, 1965. 197-200
3 Dörrie 127.
4
Eric W Weisstein, “Alhazen’s Billiard Problem”. Mathworld. Dated 1999. Viewed February 25 2006. Alexander Zouev
000051 - 060 - 4 - triangle whose legs pass through two given points inside the circle”. 5 My primary reason for choosing to investigate this focus question is that the I.B Higher Level
000051 - 060 - 3 -
Extended Essay – Mathematics
Alhazen’s Billiard Problem
Introduction:
Regarded as one of the classic problems from two dimensional geometry,
Alhazen’s Billiard Problem has a truly rich history. The problem is believed to have been first introduced by Greek astronomer Ptolemy back in 150 AD1 and then eventually noticed by 17th century Arabic mathematician Abu Ali al Hassan ibn Alhaitham (whose name was later Latinized into Alhazen)2
.
Alhazen made reference to this problem in one of his published works entitled
Optics and presented it in the form, “Find the given point on a spherical mirror at which a ray of light coming from a given point must strike in order to be reflected” 3. Nowadays, this problem is often referred to as the “Billiard Problem” because it involves locating the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in order to carom (bounce) once off the edge and strike another ball at a second given point.4 The focus question of this extended essay will be: Heinrich Dörrie also described the problem as “find in a given circle an isosceles On a circular billiards table there are two balls; at what point along the circumference must one be aimed at in order for it to strike the other after rebounding off the edge?
1
Jack Klaff, “The World May be Divided into Two Types of People – Alhazen’s Billiard Problem.”
Viewed 19 February 2005.
2
Heinrich Dörrie, 100 Great Problems of Elementary Mathematics: Their History and Solutions. Dover
Publications New York, 1965. 197-200
3 Dörrie 127.
4
Eric W Weisstein, “Alhazen’s Billiard Problem”. Mathworld. Dated 1999. Viewed February 25 2006. Alexander Zouev
000051 - 060 - 4 - triangle whose legs pass through two given points inside the circle”. 5 My primary reason for choosing to investigate this focus question is that the I.B Higher Level