satellites
![matthen:
The bending of a light ray depends only on the angle it hits the lens, and not on the thickness of the glass. This means thick lenses can be made thinner, as shown here. This is called a Fresnel lens, and is used to make magnifying sheets as well as in lighthouses. [more] [video] [code]](http://24.media.tumblr.com/f526b7f6869b62793bddfd36d9928d14/tumblr_mmulkgWsmU1qfg7o3o1_400.gif)
Org: an Emacs Mode for Notes, Planning, and Authoring
![nldmut:
christinetheastrophysicist:
Physicists Discover 13 New Solutions to Three-Body Problem
It’s the sort of abstract puzzle that keeps a scientist awake at night: Can you predict how three objects will orbit each other in a repeating pattern? In the 300 years since this “three-body problem” was first recognized, just three families of solutions have been found. Now, two physicists have discovered 13 new families. It’s quite a feat in mathematical physics, and it could conceivably help astrophysicists understand new planetary systems.
Read More.
Images/animations:Three-Body Gallery
Excerpts from the paper:Three Classes of Newtonian Three-Body Planar Periodic Orbits
Here we report the results of our ongoing numerical search for periodic collisionless planar solutions with zero-angular-momentum in a two-parameter subspace of (the full four-dimensional space of) scaled zero-angular momentum initial conditions. This subspace is defined as that of collinear configurations with one body exactly in the middle between the other two, with vanishing angular momentum and vanishing time derivative of the hyper- radius at the initial time. …
… Montgomery [21] noticed the connection between the “fundamental group of a two sphere with three punctures,” i.e., the “free group on two letters” ( a,b ), and the conjugacy classes of the “projective coloured or pure braid group” of three strands PB 3 . Graphically, this method amounts to classifying closed curves according to their topologies on a sphere with three punctures. A stereographic projection of this sphere onto a plane, using one of the punctures as the “north pole” effectively removes that puncture to infinity, and reduces the problem to one of classifying closed curves in a plane with two punctures. That leads to the aforementioned free group on two letters ( a,b ), where (for definiteness) a denotes a clockwise full turn around the right-hand-side puncture, and b denotes the counterclockwise full turn around the other puncture](http://24.media.tumblr.com/42ff73ba6ca922ad140e9d68ad93e783/tumblr_mjoc8bOFM71qdm6myo1_500.jpg)
Physicists Discover 13 New Solutions to Three-Body Problem
It’s the sort of abstract puzzle that keeps a scientist awake at night: Can you predict how three objects will orbit each other in a repeating pattern? In the 300 years since this “three-body problem” was first recognized, just three families of solutions have been found. Now, two physicists have discovered 13 new families. It’s quite a feat in mathematical physics, and it could conceivably help astrophysicists understand new planetary systems.
Images/animations:
Three-Body GalleryExcerpts from the paper:
Three Classes of Newtonian Three-Body Planar Periodic OrbitsHere we report the results of our ongoing numerical search for periodic collisionless planar solutions with zero-angular-momentum in a two-parameter subspace of (the full four-dimensional space of) scaled zero-angular momentum initial conditions. This subspace is defined as that of collinear configurations with one body exactly in the middle between the other two, with vanishing angular momentum and vanishing time derivative of the hyper- radius at the initial time. …
… Montgomery [21] noticed the connection between the “fundamental group of a two sphere with three punctures,” i.e., the “free group on two letters” ( a,b ), and the conjugacy classes of the “projective coloured or pure braid group” of three strands PB 3 . Graphically, this method amounts to classifying closed curves according to their topologies on a sphere with three punctures. A stereographic projection of this sphere onto a plane, using one of the punctures as the “north pole” effectively removes that puncture to infinity, and reduces the problem to one of classifying closed curves in a plane with two punctures. That leads to the aforementioned free group on two letters ( a,b ), where (for definiteness) a denotes a clockwise full turn around the right-hand-side puncture, and b denotes the counterclockwise full turn around the other puncture
Mind-Melter of the Day
It turns out that if you divide 1 by 998,001 you get all three-digit numbers from 000 to 999 in order.
Except for 998.
(via Futility Closet)
n : string(n) in string(n^2)
1 - is prime: True - n: 1 - n^2: 1
2 - is prime: True - n: 5 - n^2: 25
3 - is prime: False - n: 6 - n^2: 36
4 - is prime: False - n: 10 - n^2: 100
5 - is prime: False - n: 25 - n^2: 625
6 - is prime: False - n: 50 - n^2: 2500
7 - is prime: False - n: 60 - n^2: 3600
8 - is prime: False - n: 76 - n^2: 5776
9 - is prime: False - n: 100 - n^2: 10000
10 - is prime: False - n: 250 - n^2: 62500
11 - is prime: False - n: 376 - n^2: 141376
12 - is prime: False - n: 500 - n^2: 250000
13 - is prime: False - n: 600 - n^2: 360000
14 - is prime: False - n: 625 - n^2: 390625
15 - is prime: False - n: 760 - n^2: 577600
16 - is prime: False - n: 1000 - n^2: 1000000
17 - is prime: False - n: 2500 - n^2: 6250000
18 - is prime: False - n: 3760 - n^2: 14137600
19 - is prime: False - n: 3792 - n^2: 14379264
20 - is prime: False - n: 5000 - n^2: 25000000
21 - is prime: False - n: 6000 - n^2: 36000000
22 - is prime: False - n: 6250 - n^2: 39062500
23 - is prime: False - n: 7600 - n^2: 57760000
24 - is prime: False - n: 9376 - n^2: 87909376
25 - is prime: False - n: 10000 - n^2: 100000000
26 - is prime: False - n: 14651 - n^2: 214651801
27 - is prime: False - n: 25000 - n^2: 625000000
28 - is prime: False - n: 37600 - n^2: 1413760000
29 - is prime: False - n: 50000 - n^2: 2500000000
30 - is prime: False - n: 60000 - n^2: 3600000000
31 - is prime: False - n: 62500 - n^2: 3906250000
32 - is prime: False - n: 76000 - n^2: 5776000000
33 - is prime: False - n: 90625 - n^2: 8212890625
34 - is prime: False - n: 93760 - n^2: 8790937600
35 - is prime: False - n: 100000 - n^2: 10000000000
36 - is prime: False - n: 109376 - n^2: 11963109376
37 - is prime: False - n: 250000 - n^2: 62500000000
38 - is prime: False - n: 376000 - n^2: 141376000000
39 - is prime: False - n: 495475 - n^2: 245495475625
40 - is prime: False - n: 500000 - n^2: 250000000000
41 - is prime: False - n: 505025 - n^2: 255050250625
42 - is prime: False - n: 600000 - n^2: 360000000000
43 - is prime: False - n: 625000 - n^2: 390625000000
44 - is prime: False - n: 760000 - n^2: 577600000000
45 - is prime: False - n: 890625 - n^2: 793212890625
46 - is prime: False - n: 906250 - n^2: 821289062500
47 - is prime: False - n: 937600 - n^2: 879093760000
48 - is prime: False - n: 971582 - n^2: 943971582724
49 - is prime: False - n: 1000000 - n^2:1000000000000
50 - is prime: False - n: 1093760 - n^2:1196310937600
51 - is prime: False - n: 1713526 - n^2:2936171352676
52 - is prime: False - n: 2500000 - n^2:6250000000000
53 - is prime: False - n: 2890625 - n^2:8355712890625
54 - is prime: False - n: 3760000 - n^2:14137600000000
55 - is prime: False - n: 4115964 - n^2:16941159649296
56 - is prime: False - n: 5000000 - n^2:25000000000000
57 - is prime: False - n: 5050250 - n^2:25505025062500
58 - is prime: False - n: 5133355 - n^2:26351333556025
59 - is prime: False - n: 6000000 - n^2:36000000000000
60 - is prime: False - n: 6250000 - n^2:39062500000000
61 - is prime: False - n: 6933808 - n^2:48077693380864
62 - is prime: False - n: 7109376 - n^2:50543227109376
63 - is prime: False - n: 7600000 - n^2:57760000000000
64 - is prime: False - n: 8906250 - n^2:79321289062500
65 - is prime: False - n: 9062500 - n^2:82128906250000
66 - is prime: False - n: 9376000 - n^2:87909376000000
67 - is prime: False - n: 10000000 - n^2:100000000000000
68 - is prime: False - n: 10050125 - n^2:101005012515625
69 - is prime: False - n: 10937600 - n^2:119631093760000
70 - is prime: False - n: 12890625 - n^2:166168212890625
71 - is prime: False - n: 25000000 - n^2:625000000000000
72 - is prime: False - n: 28906250 - n^2:835571289062500
73 - is prime: False - n: 37600000 - n^2:1413760000000000
74 - is prime: False - n: 48588526 - n^2:2360844858852676
75 - is prime: False - n: 50000000 - n^2:2500000000000000
76 - is prime: False - n: 50050025 - n^2:2505005002500625
77 - is prime: False - n: 60000000 - n^2:3600000000000000
78 - is prime: False - n: 62500000 - n^2:3906250000000000
79 - is prime: True - n: 66952741 - n^2:4482669527413081
80 - is prime: False - n: 71093760 - n^2:5054322710937600
81 - is prime: False - n: 76000000 - n^2:5776000000000000
82 - is prime: False - n: 87109376 - n^2:7588043387109376
83 - is prime: False - n: 88027284 - n^2:7748802728416656
84 - is prime: False - n: 88819024 - n^2:7888819024312576
85 - is prime: False - n: 89062500 - n^2:7932128906250000
86 - is prime: False - n: 90625000 - n^2:8212890625000000
87 - is prime: False - n: 93760000 - n^2:8790937600000000
88 - is prime: False - n: 100000000 - n^2:10000000000000000
89 - is prime: False - n: 105124922 - n^2:11051249225506084
90 - is prime: False - n: 109376000 - n^2:11963109376000000
91 - is prime: False - n: 128906250 - n^2:16616821289062500
92 - is prime: True - n: 146509717 - n^2:21465097175420089
93 - is prime: False - n: 177656344 - n^2:31561776563446336
94 - is prime: False - n: 200500625 - n^2:40200500625390625
95 - is prime: False - n: 212890625 - n^2:45322418212890625
96 - is prime: False - n: 250000000 - n^2:62500000000000000
97 - is prime: False - n: 250050005 - n^2:62525005000500025
98 - is prime: False - n: 289062500 - n^2:83557128906250000
99 - is prime: False - n: 370156212 - n^2:137015621282188944
100 - is prime: False - n: 376000000 - n^2:141376000000000000
…
OpenSurfaces is a large database of annotated surfaces created from real-world consumer photographs. Our annotation framework draws on crowdsourcing to segment surfaces from photos, and then annotate them with rich surface properties, including material, texture and contextual information.
