Thursday, January 23, 2014

Doctor Who faves

(This is a work in progress. I'll update it as I have time.)

My favorite Doctor Who episodes in chronological order:

Ninth Doctor:

  • Dalek. Christopher Eccleston was the perfect "dark" Doctor. What the ravages of war can do to a person.
  • Father's Day. What if? A touching story about a girl and her father.
  • The Empty Child/The Doctor Dances. Great story. Loads of fun. First intro to Captain Jack.

Tenth Doctor:

  • The Christmas Invasion. Still the best of the Christmas specials. But I liked A Christmas Carol and The Snowmen well enough to include them among my favorites too.
  • School Reunion. Wherein I begin to understand how the past informs the present.
  • The Girl in the Fireplace. A wonderful love story.
  • Love & Monsters. Hilarious spoof of geek fandom.
  • Army of Ghosts/Doomsday. Heartbreaking. Rose!
  • Smith and Jones.
  • Gridlock.
  • Human Nature/The Family of Blood.
  • Blink.
  • Utopia/The Sound of Drums/Last of the Time Lords.
  • Silence in the Library/Forest of the Dead. Our first introduction to Professor River Song, and it was beautiful and sad. I wish they hadn't made River Song Rory and Amy's daughter, an assassin, and so forth though.
  • Midnight. The single scariest episode of Doctor Who. More so than Blink. The "monster" is never seen. The Doctor doesn't know what the monster is. The Doctor doesn't know what to do about the monster. Yet the real monster is humanity unmasked. Our evil nature which turns against others. A suspenseful bottle episode.
  • Turn Left. Possible worlds.
  • The Next Doctor.
  • The Waters of Mars.

Eleventh Doctor:

  • The Eleventh Hour.
  • Amy's Choice.
  • Vincent and the Doctor. The relationship between genius and madness. The blessing and curse of being able to see the invisible, of being able to see what others cannot see. The black dog of depression.
  • The Lodger. Hilarious. As is Closing Time.
  • The Pandorica Opens/The Big Bang.
  • A Christmas Carol.
  • The Impossible Astronaut/Day of the Moon. Love the G-Men, the UFOs, the Silence, the conspiracies, Nixon, the Doctor's death.
  • The Doctor's Wife.
  • The Girl Who Waited.
  • Closing Time.
  • The Wedding of River Song.
  • The Doctor, the Widow, and the Wardrobe.
  • Asylum of the Daleks.
  • The Angels Take Manhattan.
  • The Snowmen.
  • Hide. From eerie to ecstatic.
  • Day of the Doctor.

Others:

  • Favorite Doctor: Ten. Although since I've only ever seen the rebooted Doctor Who series (2005), I'm only judging from three Doctors. I like the other two Doctors though. Christopher Eccleston was great, but not around long enough. While Matt Smith had the unenviable task of following in David Tennant's footsteps, though I think Smith ended up doing a good job. Smith's acting progressively improved over time. His main flaw was his lack of gravitas for the role, or so I felt.
  • Favorite Companion: It's hard not to say Rose. Although they shouldn't have brought her back after Doomsday. That marred her status a bit. I thought Amelia Pond as a little girl meeting the Doctor in The Eleventh Hour and her relationship with him beyond that point had a lovely fairy tale like quality to it. But the way the writers dispensed with her (and Rory) was a shame. Maybe Captain Jack wins. Or does Wilfred Mott count?
  • Favorite Episode: Human Nature/The Family of Blood. This post sums it up pretty well.

Saturday, January 11, 2014

Fractal dimensions

Let's talk about fractal dimensions. Just for fun.

1. As many know, fractals are a type of geometric object which consists of self-similar patterns on all scales of magnification.

2. What's not as often discussed is fractal dimensions.

a. We know a point has zero dimensions. A line has one dimension (length). A square has two dimensions (length, width). A cube has three dimensions (length, width, depth). Likewise, fractals have fractal dimension.

b. An example that's sometimes used to illustrate fractal dimension is the Sierpinski triangle. What is a Sierpinski triangle?

Start with an equilateral triangle:

Cut out a central triangle as follows:

Next, cut out three smaller triangles as follows:

And then repeat for each of the triangles that haven't been cut out as follows:

If we continue, the result will be a Sierpinski triangle like this one:

c. Normally, if we double the lengths of a triangle, its area will be quadrupled. Say we have a triangle with base 2 and height 3. Its area would be (2 x 3) / 2 = 3. If we double its base and height, then we would have (4 x 6) / 2 = 12. And 12 is 4x greater than 3.

d. Strangely, though, if we double the lengths of a Sierpinski triangle, its area will not be quadrupled. Instead, its area will only be tripled.

This is due to the fact that when we take three smaller Sierpinski triangles to form a larger Sierpinski triangle, the larger Sierpinski triangle has doubled in size.

Thus, the Sierpinski triangle has a dimension = log 3 / log 2. That is, approximately 1.585 dimensions (if we round up).

How can we have 1.585 dimensions though? Since we cut out triangles within triangles over and over again, and in fact we could do so an infinite number of times, the resulting Sierpinski triangle is more like an assembly of points (with zero dimensions) or a grid of lines (with 1 dimension). Hence overall the Sierpinski triangle isn't 2 dimensional like a normal triangle would be, but it has 1.585 dimensions.

e. BTW, Sierpinski triangles can also be Sierpinski pyramids:

3. A few further notes:

a. In reality, a dimension either is or isn't. We can't have partial dimensions. Thus, 1.585 (or whatever) dimensions is not reality, per se, but a mathematical abstraction.

b. That said, say we have a piece of string. Say we assume this string is 1 dimensional. (Of course, all physical objects in our universe are actually at least three dimensional.) Say it's a long piece of string. Say we roll this string up so it becomes a ball. As such, the originally 1 dimensional string has become in a sense 3 dimensional.

Take another example. Say we assume a piece of paper is 2 dimensional. Say we crumple up this piece of paper. As such, the originally 2 dimensional piece of paper is 3 dimensional.

Fractals can likewise be "crumpled," so to speak.

c. Of course, the real universe in which we live is 3+1 dimensions. That is, 3 space dimensions + 1 time dimension.

We can use fractals and fractal dimensions to better understand our own world. There's a lifetime of study in this alone.

However, perhaps we can likewise use fractals and fractal dimensions to help us imagine worlds beyond our own? That is, like Edwin Abbott did in Flatland, or Ian Stewart did in Flatterland, it's possible to imagine that we live in 4+1 dimensions, for example. Perhaps fractals and fractal dimensions can be an aid here.

I'd like to write more about this in the future, time permitting.

Thursday, January 23, 2014

Doctor Who faves

(This is a work in progress. I'll update it as I have time.)

My favorite Doctor Who episodes in chronological order:

Ninth Doctor:

  • Dalek. Christopher Eccleston was the perfect "dark" Doctor. What the ravages of war can do to a person.
  • Father's Day. What if? A touching story about a girl and her father.
  • The Empty Child/The Doctor Dances. Great story. Loads of fun. First intro to Captain Jack.

Tenth Doctor:

  • The Christmas Invasion. Still the best of the Christmas specials. But I liked A Christmas Carol and The Snowmen well enough to include them among my favorites too.
  • School Reunion. Wherein I begin to understand how the past informs the present.
  • The Girl in the Fireplace. A wonderful love story.
  • Love & Monsters. Hilarious spoof of geek fandom.
  • Army of Ghosts/Doomsday. Heartbreaking. Rose!
  • Smith and Jones.
  • Gridlock.
  • Human Nature/The Family of Blood.
  • Blink.
  • Utopia/The Sound of Drums/Last of the Time Lords.
  • Silence in the Library/Forest of the Dead. Our first introduction to Professor River Song, and it was beautiful and sad. I wish they hadn't made River Song Rory and Amy's daughter, an assassin, and so forth though.
  • Midnight. The single scariest episode of Doctor Who. More so than Blink. The "monster" is never seen. The Doctor doesn't know what the monster is. The Doctor doesn't know what to do about the monster. Yet the real monster is humanity unmasked. Our evil nature which turns against others. A suspenseful bottle episode.
  • Turn Left. Possible worlds.
  • The Next Doctor.
  • The Waters of Mars.

Eleventh Doctor:

  • The Eleventh Hour.
  • Amy's Choice.
  • Vincent and the Doctor. The relationship between genius and madness. The blessing and curse of being able to see the invisible, of being able to see what others cannot see. The black dog of depression.
  • The Lodger. Hilarious. As is Closing Time.
  • The Pandorica Opens/The Big Bang.
  • A Christmas Carol.
  • The Impossible Astronaut/Day of the Moon. Love the G-Men, the UFOs, the Silence, the conspiracies, Nixon, the Doctor's death.
  • The Doctor's Wife.
  • The Girl Who Waited.
  • Closing Time.
  • The Wedding of River Song.
  • The Doctor, the Widow, and the Wardrobe.
  • Asylum of the Daleks.
  • The Angels Take Manhattan.
  • The Snowmen.
  • Hide. From eerie to ecstatic.
  • Day of the Doctor.

Others:

  • Favorite Doctor: Ten. Although since I've only ever seen the rebooted Doctor Who series (2005), I'm only judging from three Doctors. I like the other two Doctors though. Christopher Eccleston was great, but not around long enough. While Matt Smith had the unenviable task of following in David Tennant's footsteps, though I think Smith ended up doing a good job. Smith's acting progressively improved over time. His main flaw was his lack of gravitas for the role, or so I felt.
  • Favorite Companion: It's hard not to say Rose. Although they shouldn't have brought her back after Doomsday. That marred her status a bit. I thought Amelia Pond as a little girl meeting the Doctor in The Eleventh Hour and her relationship with him beyond that point had a lovely fairy tale like quality to it. But the way the writers dispensed with her (and Rory) was a shame. Maybe Captain Jack wins. Or does Wilfred Mott count?
  • Favorite Episode: Human Nature/The Family of Blood. This post sums it up pretty well.

Saturday, January 11, 2014

Fractal dimensions

Let's talk about fractal dimensions. Just for fun.

1. As many know, fractals are a type of geometric object which consists of self-similar patterns on all scales of magnification.

2. What's not as often discussed is fractal dimensions.

a. We know a point has zero dimensions. A line has one dimension (length). A square has two dimensions (length, width). A cube has three dimensions (length, width, depth). Likewise, fractals have fractal dimension.

b. An example that's sometimes used to illustrate fractal dimension is the Sierpinski triangle. What is a Sierpinski triangle?

Start with an equilateral triangle:

Cut out a central triangle as follows:

Next, cut out three smaller triangles as follows:

And then repeat for each of the triangles that haven't been cut out as follows:

If we continue, the result will be a Sierpinski triangle like this one:

c. Normally, if we double the lengths of a triangle, its area will be quadrupled. Say we have a triangle with base 2 and height 3. Its area would be (2 x 3) / 2 = 3. If we double its base and height, then we would have (4 x 6) / 2 = 12. And 12 is 4x greater than 3.

d. Strangely, though, if we double the lengths of a Sierpinski triangle, its area will not be quadrupled. Instead, its area will only be tripled.

This is due to the fact that when we take three smaller Sierpinski triangles to form a larger Sierpinski triangle, the larger Sierpinski triangle has doubled in size.

Thus, the Sierpinski triangle has a dimension = log 3 / log 2. That is, approximately 1.585 dimensions (if we round up).

How can we have 1.585 dimensions though? Since we cut out triangles within triangles over and over again, and in fact we could do so an infinite number of times, the resulting Sierpinski triangle is more like an assembly of points (with zero dimensions) or a grid of lines (with 1 dimension). Hence overall the Sierpinski triangle isn't 2 dimensional like a normal triangle would be, but it has 1.585 dimensions.

e. BTW, Sierpinski triangles can also be Sierpinski pyramids:

3. A few further notes:

a. In reality, a dimension either is or isn't. We can't have partial dimensions. Thus, 1.585 (or whatever) dimensions is not reality, per se, but a mathematical abstraction.

b. That said, say we have a piece of string. Say we assume this string is 1 dimensional. (Of course, all physical objects in our universe are actually at least three dimensional.) Say it's a long piece of string. Say we roll this string up so it becomes a ball. As such, the originally 1 dimensional string has become in a sense 3 dimensional.

Take another example. Say we assume a piece of paper is 2 dimensional. Say we crumple up this piece of paper. As such, the originally 2 dimensional piece of paper is 3 dimensional.

Fractals can likewise be "crumpled," so to speak.

c. Of course, the real universe in which we live is 3+1 dimensions. That is, 3 space dimensions + 1 time dimension.

We can use fractals and fractal dimensions to better understand our own world. There's a lifetime of study in this alone.

However, perhaps we can likewise use fractals and fractal dimensions to help us imagine worlds beyond our own? That is, like Edwin Abbott did in Flatland, or Ian Stewart did in Flatterland, it's possible to imagine that we live in 4+1 dimensions, for example. Perhaps fractals and fractal dimensions can be an aid here.

I'd like to write more about this in the future, time permitting.