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Archive: November 23, 2003

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More Cubic Bisection

Sunday,  11/23/03  04:07 PM

The other day I asked:

  1. Can a cube be sectioned in such a way as to create a regular pentagon?
  2. It appears the regular hexagonal section has the greatest area of all possible sections.  Can you prove it?

I know you've been breathlessly waiting for the answers, so here you go.

First, no, a cube cannot be sectioned to create a regular pentagon.  The closest you can do is this:

pentagonal cubic bisection

This is a "full house" pentagon; three of the sides are the same length, and the other two sides are the same length as each other, but longer than the other three.

Note: it is not necessary that one of the pentagon's vertices be coincident with a vertex of the cube.

Second, the regular hexagon is not the section with the greatest area.  I didn't mean for this to be a trick question, but I guess it was.  The section with the greatest area is this one:

maximal cubic bisection

The diagonal of each face is √2

Area: √2 = 1.41

Here's the regular hexagon again:

hexagonal cubic bisection

The diagonal of each face is √2
Each side of the hexagon is √2/2
Triangles are equilateral with area √3/8

Area: ¾√3 = 1.30

There are some other candidates as well.  In the two figures above, consider rotating the section about the dashed line as an axis.  That yields the following section (a diamond, not a square):

diamondal cubic bisection

Each side is √5/2
One chord is √2, the other is √3

Area: ½√2√3 = 1.22

And continuing the rotation, this section, a square with the minimum area of any section which passes through the center of the cube:

square cubic bisection

Section same as cube face

Area: 1

Another interesting section is this one, the largest triangular section:

triangular cubic bisection

The diagonal of each face is √2
Triangles are equilateral with area √3/8

Area: ½√3 = 0.87

Finally, here's today's bonus question:

  • What is the area of the "full house" pentagonal section?

 

Sunday,  11/23/03  11:27 PM

Citizen Smash summarizes Arnold's first week in office:

In his first official action as governor, he issued an executive order rolling back the tripling of California’s automobile registration tax shortly after taking office.  Next, he called the state legislature into special session to work on reforming the state’s workers’ compensation system; to draft a bond issue that would restructure the state’s debts (that would be placed on the March ballot); to draft a Constitutional amendment capping government spending (also for the March ballot); and to cut state spending to counter the lost revenue from vehicle licensing fees.  He also suspended all pending regulations for a 90-day executive review.

When Arnold ran into some friction from the Democrat-dominated legislature, he took to the radio airwaves across the state and warned of “severe casualties” in next year’s elections if lawmakers fight his efforts.

There’s a new sheriff in town.

Excellent!  So far, so good.

So, where were you?  Of all the things that have that "I remember when I heard the news" quality, President Kennedy's assassination is at the top of most people's list.  I was five, and I can remember sitting next to my Mom, listening to the radio.  I didn't know what was going on, but I could tell from her reaction that it wasn't good.  The only other thing that has that quality for me was the Challenger exploding.  I'm sure someday 9/11 will be on the list, too, but right now it still seems too recent.

I don't often link James Lileks (although I love his blog), but check this out, as he flips off Salam Pax.  In the same post he disses Michael Moore and bemoans the network's emphasis on the sordid Michael Jackson affair.  I cheer him on from the sidelines.

Hayling by LynnFox for FC KahunaXeni Jardin pointed out this awesome video by LynnFox for FC Kahuna's song "Hayling".  If you have a broadband connection, you must watch this.  Now.

You might have heard that CNet bought MP3.com?  They had planned to dump the extensive MP3.com archives.  Michael Robertson, founder of MP3.com (who is no longer affiliated with the company), wrote an impassioned plea that CNet turn over the MP3.com archives to archive.orgWhich they've done!  All's well that ends better.

Scoble does some introspecting about Microsoft's brand image.  "Come on, admit it.  Microsoft doesn't have the best brand in the business."  Robert, you got me.  I admit it.

Diego Doval takes a walk down memory lane.  Longhorn is either the fourth or fifth time Microsoft has tried for an object-oriented file system, depending on who's counting.

If you haven't tried X1, you should.  It provides fast file and email searching today.  It does not use metadata of any kind, rather, it does full-text pre-indexing of file contents.  And man does it scream.

Amid the debate about Longhorn, Microsoft's upcoming new version of Windows, it is worth noting that MS Office 2003 seems to have issues.  I haven't loaded it myself; there doesn't seem to be a compelling reason to upgrade...

n-generate spacefarmHey, here's something cool; n-generate.  "Graphic Design the easy way.  If you can press a button, you can n-generate."  [ via Ottmar Liebert, who goes a little crazy n-generating album covers 1, 2, 3, 4 ]

 

 
 

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