WEB-REPORT | |||||||||||

Authentication: We declare that the web-report is our own work and does not contain plagiarised material | |||||||||||

Group | 8-16 | ||||||||||

Project Title | THE INVESTIGATION OF THE PAPPUS CHAIN THEOREM THROUGH INVERSION | ||||||||||

Synopsis | The Pappus Chain is a geometrical construction, consisting of a chain of circles tangent to 2 semicircles in an arbelos, except for the first inscribed circle, which is tangent to all 3 semicircles. The Pappus Chain Theorem states that the height from the base of the Pappus Chain to the centre of a circle iCn is the product of the diameter of the nth inscribed circle the integer n. By using inversion, the proof of the Pappus Chain Theorem, which was initially proven using Euclidean Geometry, has been improved. An inversion of the Pappus Chain would produce a chain of circles inscribed within 2 parallel lines, one of which would be tangent to the reference circle, stretching to infinity. Each circle would be tangent to the adjacent circle. Right-angled triangles could be constructed and by similar triangles, the Pappus Chain Theorem could be proven. Applying similar concepts, it was proven that there are infinitely many regular Steiner Chains, and thus there are infinitely many Steiner Chains that can exist given 2 non-intersecting circles. After extending the study to 3-dimensions and applying inversion concepts in 3-dimensions in a hexlet, it was proven that in a hexlet, there must be 6 spheres surrounding the central sphere(s). Using a suitable software, and by taking a reference point on the globe and inverting the globe about the reference point, an azimuthal equidistant projection can be easily produced. | ||||||||||

Link to start page | Click HERE to access web-report | ||||||||||

Special
instructions for evaluator to take note |
Nil | ||||||||||

Team Members (Names & Classes) |
Group Leader: | ||||||||||

SCOTT LOO YI HE 4S1 | |||||||||||

Group Member/s: | |||||||||||

NG KAI JIE 4S1 | |||||||||||

SUN LONGXUAN 4S1 | |||||||||||

TAN ZHI HENG 4S2 | |||||||||||