Transcript Brilliant-Cut Diamonds and Other Tricks of the

Brilliant-Cut Diamonds and Other
Tricks of the Light
John D Barrow
Diamond Geezer
Marcel Tolkowsky (1899-1991)
Diamond Design: A study of the reflection and refraction of light in a diamond,
E & F.N. Spon, Ltd., London, 1919, while a graduate student at Imperial College
Old Diamond Cuts
Stylistic evolution from the most primitive (Point Cut)
to the most advanced pre-Tolkowsky cut (Old European)
Brilliant Cut Diamonds
MT: “We have therefore to study these optical properties in order to be
able to apply them to the problem we have now before us :
the calculation of the shape and proportions of a perfectly cut diamond.”
Reflection of Light
A
B
a a
C
Minimises the time to traverse A  C  B
mirror
lifeguard
The Maths of Baywatch
beach
shoreline
sea
swimmer
is the least time path
Refraction of Light
n = sin(θ1)/sin(θ2) = (speed in medium 1)/(speed in medium 2)
The least time path
A Fish-eye View
Critical angle for air-water interface is sin-1(1.00/1.33) = 48.8 deg
n(diamond) = sin(i)/sin(r) = (speed in vacuum)/speed in diamond)
n(diamond) = sin(iair)/sin(rdiamond) = 2.42
Critical angle with air = sin-1(1/n)
n(air) = 1.0003  1 , n(water) = 1.33, n(glass) = 1.52
Crit(diamond) = 24.4 deg, Crit(glass) = 42 deg, Crit(water) = 48.8 deg
Light speed in vacuum = 186,282 miles/second = 3 x 108 m/sec
Light speed in diamond = 77,056 miles/second = 1.24 x108 m/sec
Any ray which strikes the inside surface on the at an angle greater
than 24.4 degrees will not escape the diamond.
‘Fire’ from Dispersion
Pink Floyd: Dark Side of the Moon
1973
1993
2003
Dispersion
is also very large
“Fire”
The ‘Goldilocks’ Design
Pavilion 40.75 deg
Crown 34.5 deg
Table Ratio 53%
“For absolute total reflection to occur at the first facet,
the inclined facets must make an angle of not less
that 48°52' with the horizontal.”
“For absolute total reflection at the second facet,
the inclined facets must make an angle
of not more than 43°43' with the horizontal.”
“For [outgoing] refraction,  may be less or more than 45°.
When more, the best value is 49°15', but it is unsatisfactory.
When less, the best value is 40°45', and is very satisfactory,
as the light can be arranged to leave with the best possible dispersion.”
“Upon consideration of the above results, we conclude that
the correct value for  is 40°45', and gives the most vivid fire
and the greatest brilliancy, and that although a greater angle
would give better reflection, this would not compensate for the
loss due to the corresponding reduction in dispersion.
In all future work upon the modern brilliant we will therefore
take  = 40°45'. “
Marcel Tolkowsky, 1919
MOVIES WITH VARYING CROWN AND PAVILION ANGLES
Wireframe
model of
diamond with
reflected
edges
Photorealistic
movies.
"Office" illumination
Photorealistic
movies.
IdealScope illumination
Ray Tracing
Crown angle varies from 30 to 40.
Pavilion angle 40.7.
Crown angle 35.
Pavilion angle varies from 36 to
46.
Crown angle varies from 31.25 to 37.75.
Pavilion angle varies from 39.15 to 41.85.
Direct dependence.
Crown angle varies from 31.2 to 37.8.
Pavilion angle varies from 41.25 to
39.75.
Inverse dependence
(OctoNus-MSU line).
OctNus: Sergey Sivovolenko, Yuri Shelementiev (2002)
MOVIES WITH TILTING DIAMONDS. CONSTANT
PROPORTIONS.
Photorealistic
movies.
"Office" illumination
Photorealistic
images.
IdealScope illumination
Ray Tracing
Crown angle 34.5.
Pavilion angle
40.5.
Crown angle 34.5.
Pavilion angle
41.5
Crown angle 32.
Pavilion angle
41.
Crown angle 33.
Pavilion angle
40.
OctNus; Sergey Sivovolenko, Yuri Shelementiev (2002)
Mirror, Mirror on the Wall
Mirror, Mirror on the Wall
Tarnowski
Alter Ego Magritte
NOT
OR
Through the Looking Glass
Angled Mirrors
45
45
30
30
60
No right-left reversal
in right-angled mirrors
Reduce the angle and left-right
reversal occurs at 60 degrees
What do you really look like?
2 lefts vs 2 rights
How big do you need to make a mirror ?
How big does a mirror have to be,
so that you can see all of your face in it ?
Angle of incidence equals angle of reflection.
Whatever the distance of the viewer, light from the top
of the head is reflected from the mirror at a point half-way
between the top or bottom of the head and the eyes.
The mirror size is half the height of your head.
The size of reflected image is the same no matter how far
you are from the image
a
a
a
b
b
b
Face size = 2a +2 b
Mirror image size = a + b
‘Invisible’ Objects
At least 4 reflections needed
Pepper’s Ghost
Invented by Henry Dircks and used by John Pepper, 1862
The Optics of Pepper’s Ghost
Gradually raise the lights
in the darkened room on
the left and the ghost
appears
Thomas Tobin’s Proteus Cabinet
London Polytechnic Institution,1865
“Here, but not Here”
Safe
zone
The first magic cabinet trick
The first trick “all done with mirrors”
Look
45 deg