# Chaitanya's Random Pages

## February 28, 2015

### Large US+Canada box office openings

Filed under: movies and TV — ckrao @ 1:38 pm

The site boxofficemojo.com lists the opening weekend grosses of movies in the US and Canada dating back to the early 1980s. Via this page on top opening weekends, I worked out movies that at their time of release attained the n’th highest grossing opening weekend where n ranges from 1 to 10 (all dollar amounts in $US). It gives a perspective on how big some movies were at the time. It also shows how movie grosses have grown through inflation and more frontloading over the years. Note that only opening weekends are shown here – for example Superman’s third weekend was once the largest grossing weekend at the time, but is not listed here. n=1 (i.e. current and previous record-breaking openings):  Title Opening Date (mm/dd/yyyy) Marvel’s The Avengers$207,438,708 5/4/2012 Harry Potter and the Deathly Hallows Part 2 $169,189,427 7/15/2011 The Dark Knight$158,411,483 7/18/2008 Spider-Man 3 $151,116,516 5/4/2007 Pirates of the Caribbean: Dead Man’s Chest$135,634,554 7/7/2006 Spider-Man $114,844,116 5/3/2002 Harry Potter and the Sorcerer’s Stone$90,294,621 11/16/2001 The Lost World: Jurassic Park $72,132,785 5/23/1997 Batman Forever$52,784,433 6/16/1995 Jurassic Park $47,026,828 6/11/1993 Batman Returns$45,687,711 6/19/1992 Batman $40,489,746 6/23/1989 Ghostbusters II$29,472,894 6/16/1989 Indiana Jones and the Last Crusade $29,355,021 5/24/1989 Beverly Hills Cop II$26,348,555 5/20/1987 Indiana Jones and the Temple of Doom $25,337,110 5/23/1984 Return of the Jedi$23,019,618 5/25/1983 Star Trek II: The Wrath of Khan $14,347,221 6/4/1982 Superman II$14,100,523 6/19/1981 Star Trek: The Motion Picture $11,926,421 12/7/1979 Every Which Way But Loose$10,272,294 12/20/1978

n=2 (i.e. the second largest opening at the time)

 Title Opening Date Iron Man 3 $174,144,585 5/3/2013 Star Wars: Episode III – Revenge of the Sith$108,435,841 5/19/2005 Shrek 2 $108,037,878 5/19/2004 The Matrix Reloaded$91,774,413 5/15/2003 Planet of the Apes (2001) $68,532,960 7/27/2001 The Mummy Returns$68,139,035 5/4/2001 Star Wars: Episode I – The Phantom Menace $64,820,970 5/19/1999 Independence Day$50,228,264 7/3/1996 Lethal Weapon 3 $33,243,086 5/15/1992 Terminator 2: Judgment Day$31,765,506 7/3/1991 Rocky III $12,431,486 5/28/1982 The Cannonball Run$11,765,654 6/19/1981 Smokey and the Bandit II $10,883,835 8/15/1980 The Empire Strikes Back$10,840,307 6/20/1980

n=3:

 Title Opening Date The Dark Knight Rises $160,887,295 7/20/2012 The Hunger Games$152,535,747 3/23/2012 The Twilight Saga: New Moon $142,839,137 11/20/2009 Shrek the Third$121,629,270 5/18/2007 Harry Potter and the Prisoner of Azkaban $93,687,367 6/4/2004 Harry Potter and the Chamber of Secrets$88,357,488 11/15/2002 Star Wars: Episode II – Attack of the Clones $80,027,814 5/16/2002 Hannibal$58,003,121 2/9/2001 Mission: Impossible II $57,845,297 5/24/2000 Toy Story 2$57,388,839 11/24/1999 Austin Powers: The Spy Who Shagged Me $54,917,604 6/11/1999 Men in Black$51,068,455 7/2/1997 The Lion King $40,888,194 6/24/1994 Rambo: First Blood Part II$20,176,217 5/22/1985 Star Trek III: The Search for Spock $16,673,295 6/1/1984 n=4:  Title Opening Date X-Men: The Last Stand$102,750,665 5/26/2006 Harry Potter and the Goblet of Fire $102,685,961 11/18/2005 X2: X-Men United$85,558,731 5/2/2003 Austin Powers in Goldmember $73,071,188 7/26/2002 Rush Hour 2$67,408,222 8/3/2001 Pearl Harbor $59,078,912 5/25/2001 Mission: Impossible$45,436,830 5/22/1996 Twister $41,059,405 5/10/1996 Back to the Future Part II$27,835,125 11/22/1989 Rocky IV $19,991,537 11/27/1985 Beverly Hills Cop$15,214,805 12/5/1984 Jaws 3-D $13,422,500 7/22/1983 Superman III$13,352,357 6/17/1983

n=5:

 Title Opening Date The Twilight Saga: Breaking Dawn Part 1 $138,122,261 11/18/2011 Iron Man 2$128,122,480 5/7/2010 Pirates of the Caribbean: At World’s End $114,732,820 5/25/2007 How the Grinch Stole Christmas$55,082,330 11/17/2000 Interview with the Vampire $36,389,705 11/11/1994 Home Alone 2: Lost in New York$31,126,882 11/20/1992 Bram Stoker’s Dracula $30,521,679 11/13/1992 Star Trek IV: The Voyage Home$16,881,888 11/26/1986 The Best Little Whorehouse in Texas $11,874,268 7/23/1982 E.T.: The Extra-Terrestrial$11,835,389 6/11/1982

n=6:

 Title Opening Date The Hunger Games: Catching Fire $158,074,286 11/22/2013 Harry Potter and the Deathly Hallows Part 1$125,017,372 11/19/2010 Alice in Wonderland (2010) $116,101,023 3/5/2010 The Passion of the Christ$83,848,082 2/25/2004 Monsters, Inc. $62,577,067 11/2/2001 X-Men$54,471,475 7/14/2000 Ace Ventura: When Nature Calls $37,804,076 11/10/1995 Robin Hood: Prince of Thieves$25,625,602 6/14/1991 Total Recall $25,533,700 6/1/1990 Teenage Mutant Ninja Turtles$25,398,367 3/30/1990 Ghostbusters $13,578,151 6/8/1984 Staying Alive$12,146,143 7/15/1983

n=7:

 Title Opening Date Transformers: Revenge of the Fallen $108,966,307 6/24/2009 Spider-Man 2$88,156,227 6/30/2004 Batman and Robin $42,872,605 6/20/1997 Lethal Weapon 2$20,388,800 7/7/1989 Star Trek V: The Final Frontier $17,375,648 6/9/1989 n=8:  Title Opening Date The Twilight Saga: Breaking Dawn Part 2$141,067,634 11/16/2012 The Lord of the Rings: The Return of the King $72,629,713 12/17/2003 Godzilla$44,047,541 5/20/1998 The Flintstones $29,688,730 5/27/1994 Gremlins$12,511,634 6/8/1984

n=9:

 Title Opening Date Finding Nemo $70,251,710 5/30/2003 The Mummy$43,369,635 5/7/1999 Deep Impact $41,152,375 5/8/1998 n=10:  Title Opening Date Toy Story 3$110,307,189 6/18/2010 Indiana Jones and the Kingdom of the Crystal Skull $100,137,835 5/22/2008 Iron Man$98,618,668 5/2/2008 Dick Tracy \$22,543,911 6/15/1990

(To create the above lists the movie lists in decreasing order of gross were pasted into Excel and the opening weekend date was converted to a number by creating a new column with formula =–TEXT(,”mm/dd/yyyy”). This was then converted to a rank by a countif formula to count the number of occurrences with higher gross that predated each movie. Finally a filter was applied to select ranks 1 to 10.)

## February 27, 2015

### Cross sections of a cube

Filed under: mathematics — ckrao @ 9:59 pm
When a plane intersects a cube there is a variety of shapes of the resulting cross section.
• a single point (a vertex of the cube)
• a line segment (an edge of the cube)
• a triangle (if three adjacent faces of the cube are intersected)
• a parallelogram (if two pairs of opposite faces are intersected – this includes a rhombus or rectangle)
• a trapezium (if two pairs of
• a pentagon (if the plane meets all but one face of the cube)
• a hexagon (if the plane meets all faces of the cube)

The last five of these (the non-degenerate cases) are illustrated below and at http://cococubed.asu.edu/images/raybox/five_shapes_800.png . Some are demonstrated in the video below too.

One can use [1] to experiment interactively with cross sections given points on the edges or faces, while [2] shows how to complete the cross section geometrically if one is given three points on the edges.

Let us be systematic in determining properties of the cross sections above. Firstly, if the plane is parallel to an edge (any of four parallel edges), the cross section can be seen to be a line or rectangle with the longer dimension of length at most $\sqrt{2}$ times the other. That rectangle becomes a square if the plane is parallel to a face.

If the plane is not parallel to a face, we may set up a coordinate system where a unit cube is placed in the first octant aligned with the coordinate axes and the normal to the plane has positive x, y and z coordinates. In other words, we may assume the plane has equation $ax + by + cz = 1$ and intercepts at $(1/a,0,0), (0,1/b,0)$ and $(0,0,1/c)$, where $a, b, c$ are positive.

The cross section satisfies $ax + by + cz = 1$ and the inequalities $0 \leq x \leq 1$, $0 \leq y \leq 1$ and $0 \leq z \leq 1$. This can be considered the intersection of the two regions

$\displaystyle ax + by + cz = 1, 0 \leq x, 0 \leq y, 0 \leq z,$

$\displaystyle ax + by + cz = 1, x \leq 1, y \leq 1, z \leq 1,$

each of which is an acute-angled triangle in the same plane (acute because one can show that the sum of the squares of any two sides is strictly greater than the square of the third side). Note that the triangles have parallel corresponding sides, being bounded by the pairs of parallel faces of the cube $x = 0, x=1, y = 0, y= 1, z=0, z=1$. Hence the two triangles are oppositely similar with a centre of similarity.

The following diagram shows the coordinates of the vertices of the two triangles, which in this case intersect in a hexagon.

The centre of similarity of the two triangles is the intersection of two lines joining corresponding sides – this can be found to be the point $(1/(a+b+c), 1/(a+b+c), 1/(a+b+c))$, which is the intersection of the unit cube’s diagonal from the origin (to $(1,1,1)$) and the plane $ax + by + cz = 1$.

Side lengths of the triangles and distances between corresponding parallel sides may be found by Pythagoras’ theorem and are shown below for one pair of corresponding sides (the remaining lengths can be found by cyclically permuting $a,b,c$).

To sum up, all of the possible cross sections of a cube where the plane is not parallel to an edge can be described by the intersection of two oppositely similar triangles with corresponding sides parallel.

The type of polygon obtained depends on which vertices of the figure below are selected, as determined by the values of $a,b,c$.

In this figure a vertex for the cross-sectional polygon is chosen if the constraint associated with it is satisfied. A red vertex has a conflicting constraint with its neighbouring two blue vertices, so either a red point or one or more blue points in this area can be chosen. Note that for the plane to intersect the cube at all we require $(1,1,1)$ to be on the different side of the plane from the origin, or in other words, $a + b + c \geq 1$.

Let us look at a few examples. Firstly, if $a, b, c$ are all greater than 1 we choose the following triangle.

Similarly if $a+b, b+c, c+a$ all are less than 1, the oppositely similar triangle on the red vertices would be chosen.

Next, if $c > 1, a < 1, a+b < 1$ we obtain the following parallelogram.

If $c > 1, a < 1, a+b > 1$ we obtain either a pentagon (parallelogram truncated at a vertex) or a trapezium depending on whether $b < 1$ or $b \geq 1$ respectively.

$b < 1$:

$b\geq 1$:

Finally, if $a, b, c$ are less than 1 and $a+b, b+c, c+a$ are greater than 1, we obtain a hexagon.

For details on calculating the areas of such polygons refer to [3], especially the method applying the area cosine principle that relates an area of a figure to its projection. For calculating volumes related to regions obtained by the cross section refer to [4].

#### References

[1] Cross Sections of a Cube: http://www.wou.edu/~burtonl/flash/sandbox.html

[2] Episode 16 – Cross sections of a cube: http://sectioneurosens.free.fr/docs/premiere/s02e16s.pdf

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