Monday, March 12, 2018

Lesson 12-6: The Fundamental Theorem of Similarity (Day 126)

Lesson 12-6 of the U of Chicago text is called "The Fundamental Theorem of Similarity." In the modern Third Edition of the text, the Fundamental Theorem of Similarity appears in Lesson 12-4.

This past weekend was the transition to Daylight Saving Time. It is my blog tradition to write about clock-related issues on the day we spring forward. Today's post will discuss two main issues:

  • State proposals for Year-Round Standard or Year-Round Daylight Saving Time
  • Adding a clock to a Calendar Reform proposal
Let's start with proposals to eliminate the biannual clock shift. The big news is in Florida:

The “Sunshine Protection Act” means that Florida would not set their clocks back in the fall, when the rest of the Eastern United States does. This change would give Florida residents more sunshine in the evening during the winter.

Many states have made similar proposals, including the Kansen Chu bill in California. But this is the first time such a bill has actually passed in a state legislature.

The bill now awaits the governor's signature -- but that's not enough for the law to have effect. Since it's a Year-Round Daylight Saving Time bill, Congress must approve of the change. Recall that officially, there's no such thing as Year-Round DST. Instead, Year-Round DST is actually Year-Round Standard Time for the next time zone. The new time zone is often known as "Atlantic Time" in parts of Canada -- and closer to Florida, Puerto Rico would also be in this time zone.

Recall that parts of New England were also considering moving to Atlantic Time. But there is one complication in Florida that doesn't apply in New England -- the fact that the Florida Panhandle currently uses Central Time, not Eastern Time. This is mentioned in the first comment posted at the link above:

James W. Schneider:
The bill needs to change. It should permanently place the entire state on Eastern Daylight Savings Time. That would place the entire state under one time zone. The way the Bill reads now, Floridia will still have two time zones, each one moved ahead one hour.

Actually, this was indeed the original proposal -- all of Florida would be on Atlantic Time. But this was strongly opposed by Florida Panhandlers. Here's a link to a newspaper from Pensacola, the major city in the Panhandle:

Before the vote, bill sponsor Greg Steube, R-Sarasota, removed a provision that would have placed all of the state in the Eastern time zone, noting that people in Northwest Florida objected to switching from Central time.

“Northern Florida people in the Central Time zone overwhelmingly want to stay where they are,” said committee Chairman Bill Montford, a Tallahassee Democrat whose sprawling 11-county district includes areas in each time zone.

Returning to the original Accuweather link, another commenter is concerned:

Rhea Ringgard:
no one thinks about the fact that the sun will rise later. There's no way to save that much light. Sorry. for a short while it will be dark until 8:00 in the east - don't know the exact time in the south. You may save an hour during October, November and December. Is it all worth it?

Recall that the original purpose of DST is to keep sunrise between 5 AM and 8 AM all year. The state of Florida is far enough south that a single Year-Round clock (with no biannual shift) might be able to keep sunrise in the desirable range, especially considering that the state is farther south than Arizona, which has Year-Round Standard Time.

Let's calculate sunrise times for three cities in the Sunshine State -- the capital Tallahassee, the Panhandle city of Pensacola, and Parkland (a city in the news lately, to represent the southern part of the state):

Assuming Year-Round Eastern Daylight/Atlantic Standard Time (the original bill)
City                    Sunrise on Jan. 11th  Sunrise on June 11th
Pensacola           8:46                           6:46
Tallahassee         8:35                           6:34
Parkland             8:21                           6:32

We notice that all of these times have sunrise after 8 AM in the winter, which is undesirable. On the other hand, let's convert all of these to Year-Round (Eastern) Standard Time instead:

Assuming Year-Round Central Daylight/Eastern Standard Time
City                    Sunrise on Jan. 11th  Sunrise on June 11th
Pensacola           7:46                           5:46
Tallahassee         7:35                           5:34
Parkland             7:21                           5:32

Sunrise for all three cities is now in the desirable range of 5-8 AM the whole year. Therefore, the best thing for Florida to do is for the Panhandle to keep the clocks forward all year, and for the rest of the state to keep the clocks back all year. The entire state would be in one time zone, except this time zone would be Eastern Standard Time -- not Atlantic Standard (Eastern Daylight) time.

My proposal fits the old Sheila Danzig plan, where the entire Central and Eastern time zones are combined into one super zone:

Notice that Sheila Danzig actually lives in Florida -- indeed, she's about 25 miles from Parkland (where the Valentine's Day massacre took place). I notice that even though Danzig no longer updates her website, she's been interviewed for recent articles discussing DST. But strangely, I see no Danzig articles dated 2018, even though now is when that a Year-Round DST plan is making real progress in her actual home state.

I wonder what Sheila would say about the current bill. I assume that she'd prefer Year-Round Standard Time for her part of Florida, but that Year-Round DST may be preferable to a biannual clock shift (which she completely opposes).

Here's what the Florida legislature should do to promote the Danzig plan. First, it should abolish DST (that is, implement Year-Round Standard Time) in all of Florida that lies in Eastern Time. This can be done without approval from Congress, and indeed this is similar to what Indiana used to do. Then the state can ask Congress to extend Eastern Time to the Panhandle. It's too late to do this elegantly in 2018, but perhaps in 2019 Eastern Florida can skip the clock change in March, then maybe before the clock change in November, Congress will have approved the time zone change for the Panhandle.

The Danzig plan can then extend to other states. Some commuters travel between Pensacola and Alabama, as well as between Tallahassee and Georgia, so these states can convert to Danzig time. In this case, Georgia should go first, since Congress isn't needed to approve the change.

The other Danzig time zone combines the Pacific and Mountain time zones. The best state to convert to Danzig time first is Utah. This is because Congress doesn't need to approve the change, and the Beehive State borders the one state (Arizona) that is already on Danzig time.

I've already proposed the name "Arizona Time" for the new West Coast time zone once the Danzig conversion is complete. The new East Coast time zone could be called "Florida Time," if the Sunshine State becomes the first state to convert to it. It could have been called "Indiana Time," except that the Hoosier State no longer observes it -- and besides, "Florida Time" honors Sheila Danzig, its original proponent. (It's also possible to name the zones Cactus Time and Grapefruit Time after the spring training leagues that play in those respective states.)

OK, let's move on to today's other topic -- adding a clock to a Calendar Reform proposal. Back on New Year's Day 2016, I wrote about a new calendar -- the Eleven Calendar. As its name implies, the number 11 figures prominently in its structure. There are 11 days in a week and 11 months in a year, and with three weeks per month, the number of days in a year works out to be 363. Either extra days or a Leap Week can be added to correct the average year length to 365.24+.

As 2017 began I wrote a little more about the Eleven Calendar, but two months ago I forgot all about it and discussed the Yerm Calendar instead, including how to celebrate holidays. But I had no hand in the creation of the Yerm Calendar, so I shouldn't try to place its holidays on it. The Eleven Calendar, meanwhile, is my own invention.

Anyway, my New Years 2016 post drew the following comment:

wendy krieger:
If you really want to go further, on an ordinary clock, the hour and minute hands cross 22 times a day, at an average of 66 minutes. It's actually pretty easy to read time from this, since 

H: the hour is the last crossed hour, at 10:15, this is 9 times.

M: this is the number of minutes after the crossing, it's 20 minutes by the current angle of the hour and minute, and add a minute for each ten minutes between the hands, so 22 m

So 10:15 comes to be 9:22.

Let's remind ourselves who this commenter is. Wendy Krieger is a member of the website Dozens Online, which discusses the base 12 numeration system. I've linked to Dozens Online before, but this board recently migrated, so let me link to the new site:

Now four years ago, Krieger mused about a calendar based on the numbers 11 and 33. (The original significance of 33 is that it's the length of the Dee solar cycle -- see my December 31st, 2017 post for more info.) Indeed, her post is what inspired me to create the Eleven Calendar -- well, that and the fact that no other calendar is based on the number 11 (while nearby primes seven and 13 figure prominently in other calendars).

So in this comment above, Krieger proposes a clock that goes with the Eleven Calendar. Just like the calendar, the clock is based on multiples of 11. So the day is divided into 22 hours, since 22 is the multiple of 11 that's closest to 24.

Now Wendy writes about hand crossings here. This refers to the old riddle, "How many times a day do a clock's hands overlap?" It's discussed at the following link:

According to the link, the hands cross 22 times -- and hey, we want 22 hours on the Eleven Clock. So in other words, every time the hands on a standard clock overlap is a new hour on the Eleven Clock.

Furthermore, the link above tells us that the hands cross every 65.45 minutes. Even though this rounds off to 65, we choose 66 instead -- because 66 is also a multiple of 11. And so if we divide the day into 22 hours and each hour into 66 minutes, each minute will be very close to a standard minute (about 59.5 seconds, as it turns out).

In Krieger's example above, she wants to convert 10:15 to the Eleven Clock. The hour works out to be 9, since the most recent crossing was during the 9 hour.

Then the minutes can be calculated by actually counting the minutes between the hands, since the minutes in both clocks are nearly equal. Notice Wendy's extra step "add a minute for each ten minutes between the hands." This accounts for the fact that the hour hand has moved since the last crossing.

It's easier to understand Krieger's algorithm if we try a few more examples:

4:02 Standard Time. The last time the hands crossed was before 4:00 -- during the 3 hour. The hour hand is basically pointing at the 4 (minute 20) and the minute hand is pointing to minute 2, so there are 42 minutes between the hands. For every 10 minutes we add an extra minute, so we need four extra minutes. This gives us 3:46 Eleven Time.

8:36 Standard Time. The last time the hands crossed was before 8:00 -- during the 7 hour. The hour hand is pointing at 3 points past the 8 (minute 43) and the minute hand is pointing to minute 36, so there are 53 minutes between the hands. For every 10 minutes we add an extra minute, so we need five extra minutes. This gives us 7:58 Eleven Time. In this example, it might be easier to count backwards from the 8 hour crossing. There are seven minutes between the hands. For every 10 minutes we add an extra minute -- and since 0.7 rounds to 1, we might add the minute anyway. This gives us 8 minutes to 8 Eleven Time, which is the same as 7:58.

8:08 Standard Time. The last time the hands crossed was before 8:00 -- during the 7 hour. The hour hand is pointing at almost 1 point past the 8 (minute 41) and the minute hand is pointing to minute 8, so there are 27 minutes between the hands. For every 10 minutes we add an extra minute, so we need three extra minutes (2.7 rounds to 3). This gives us 7:30 Eleven Time.

3:48 Standard Time. This last time the hands crossed was after 3:00 -- during the 3 hour. The hour hand is pointing at 4 points past the 3 (minute 19) and the minute hand is pointing to minute 48, so there are 29 minutes between the hands. For every 10 minutes we add an extra minute, so we need three extra minutes (2.9 rounds to 3). This gives us 3:32 Eleven Time. In one more minute, the hands will be almost directly opposite one another. This gives us half past 3 Eleven Time, which is the same as 3:33.

Try 6:45 Standard Time on your own. You should obtain 6:12 Eleven Time.

We can check these by looking at the list at the above link. For example, the list gives the time of the 7 crossing as 7:38:10. Sure enough, 8:08 is thirty minutes after this crossing and 8:36 is 58 minutes after this cross, so the Eleven Clock times 7:30 and 7:58 are correct.

It might be interesting to add seconds to the Eleven Clock. But notice that, for example, the 1 crossing is at 1:05:27 -- when the hour and minute hands cross, the second hand is far away. Of course, at around 1:05:05 the minute and second hands overlap, but the hour hand is so close to the other two hands that all three appear to coincide. (But the three hands coincide exactly only at noon and midnight.) Anyway, there is so much rounding in the above algorithm that it would be pointless to calculate seconds. Still, we can imagine that there are 66 seconds in each minute. Then each second is about 0.9 SI seconds in length.

Some people might point out that the Krieger algorithm is needlessly complicated. We know that telling time to the minute is a third grade Common Core Standard. But many students well past the third grade consider analog clocks to be obsolete and make no effort to remember how to tell analog time forever.

Of course, the truth is that in a world that uses the Eleven Clock, there would only be eleven hours on a clock, and six points between each hour to mark the 66 minutes. The Krieger algorithm merely tells us how to take advantage of crossings to convert time on a standard (Twelve) clock to Eleven. And it's because of crossings that it's actually easier to convert analog time to Eleven than digital time -- it might be hard to see why 3:49 converts to 3:33, but on an analog clock it's obvious that the hands are opposite one other.

Notice that in an Eleven world, quarter hours might be awkward since 4 doesn't divide 66. Perhaps third hours (22 minutes) might be more commonly known in this world. It's a shame, though, since quarter hours are easier to discern (right angles between the hands).

Finally, an Eleven world might have a base 11 (or undecimal) system. At the Dozens Online website, 
the symbol {a} denotes decimal and {c} denotes dozenal, so presumably {b} is undecimal.

{b} (default undecimal)

On the proposed calendar, a year contains 10 months, a month contains 3 weeks, a week contains 10 days, a day contains 20 hours, an hour contains 60 minutes, and a minute contains 60 seconds.

Also, notice that 100 is one more than Krieger's favorite base, "twelfty" (the long hundred).

{a} (default decimal)

Of course, base 11, like most odd bases, is not commonly used at Dozens Online. But the following link mentions how to measure time in all bases from five to 16, plus selected high even bases (including twelfty):

Here the undecimal "timel" is given as 11^-6 day, or about 0.05 seconds. This is clearly different from the Eleven Clock "second" we calculated earlier. The "timel" was calculated using pure undecimal divisions of the day, while our clock uses numbers such as 22 and 66, not just 11. It's possible that native undecimalists might use 66 as an "auxiliary base" and thus create a clock more like our Eleven Clock than the undecimal "timel."

By the way, suppose we have an actual clock with 11 hours and 66 minutes, so we can read the time on the Eleven Clock directly. My question is, how can we convert this time back to the Standard Twelve Clock? Watching when the hands cross doesn't work because the hands meet only ten times per half-day, so this would give us a Ten Clock, not the standard Twelve Clock.

The answer is that we must use reflections. We reflect the minute hand across the vertical line that marks noon/midnight. If the reflection image of the minute hand crosses the hour hand, then it's an exact hour on the Twelve Clock. Everything else is also opposite -- we must round the hour of the crossing up to the next hour instead of down. We must count the minutes between the hour and reflected minute hand counterclockwise. And we must subtract one minute for every eleven minutes in this difference. The conversion from Eleven to Twelve is thus more complicated. (Of course, if we tried this backwards algorithm on the standard Twelve Clock we end up with a Thirteen Clock.)

There's one more thing I want to discuss about the Eleven Clock. How would time zones and Daylight Saving Time work on the new clock, anyway?

The most straightforward thing to do is to have 22 time zones instead of 24, one for each hour on the new clock. The width of each time zone would be 360/22 = 16.36 degrees of longitude, rather than the usual 15 degree.

Now there's no reason for any of these time zones to line up with the old zones at all. For example, the old International Date Line is supposed to be at longitude 180. But it has been shifted -- for example, the date is different in Samoa and American Samoa. This is so that Samoa can be on the same side of the line as Australia and Asia, while American Samoa is on the same side of the line as the United States.

The Samoa/American Samoa problem means that any attempt to name a simple longitude for the International Date Line is doomed. But how about this -- let's give the IDL as 171 degrees west, which passes near Samoa/American Samoa -- then those islands can then decide which side of the line they want to be on. Then the 22 time zones are given as follows:

171.81W - 155.45W 1 AM
155.45W - 139.09W 2 AM
139.09W - 122.73W 3 AM
122.73W - 106.36W 4 AM
106.36W - 90.00W 5 AM
90.00W - 73.64W 6 AM
73.64W - 57.27W 7 AM
57.27W - 40.91W 8 AM
40.91W - 24.55W 9 AM
24.55W - 8.18W 10 AM
8.18W - 8.18E noon
8.18E - 24.55E 1 PM
24.55E - 40.91E 2 PM
40.91E - 57.27E 3 PM
57.27E - 73.64E 4 PM
73.64E - 90.00E 5 PM
90.00E - 106.36E 6 PM
106.36E - 122.73E 7 PM
122.73E - 139.09E 8 PM
139.09E - 155.45E 9 PM
155.45E - 171.81E 10 PM
171.81E - 171.81W midnight

As usual, the center longitude of each time is used as a base -- noon follows the mean sun at that particular longitude. Actually, I changed the IDL from 171W to 171.81W so that one of the time zones can be centered at longitude 0 (Greenwich).

Notice what happens in the United States, though. Instead of four time zones, the lower 48 states are now mostly squeezed into three time zones (listed as 4, 5, 6 AM above). Only parts of either coast are in the 3 or 7 AM time zones (starting from just north of San Francisco on the West Coast and New York City/Long Island on the East Coast, up to the Canadian border). Most likely, the time zones would be fudged so that the entire continental U.S., including Washington, Oregon, and New England, can fit inside three time zones.

Meanwhile in Canada, those currently on Atlantic Time might choose the 7 AM time zone, and parts of Newfoundland (currently 90 minutes ahead of New York) might even choose 8 AM.

The new time zone borders are at 90W and 106.36W. It's easy to determine the states that are on the 90W line by using the Degree Confluence website:

Let's call the 90W-73.64W time zone "Eastern Time" and 106.36W-90W zone "Central Time." So this is the list of states that would be on the border between Eastern and Central Time:

Louisiana, Mississippi, Tennessee, Missouri, Illinois, Wisconsin, Michigan, Minnesota

The states of Mississippi and Wisconsin are split by 90W, as is the Canadian province of Ontario, while the other states are clearly on one side or the other. New Orleans is close to the line but, like most of Louisiana, is in Central Time. But Memphis is also on the Central Time side, while the rest of Tennessee is on the Eastern Time side.

Sometimes the Mississippi River is suggested as a time zone line. So states east of the river would be in Eastern Time, while states to its west are in Central Time.

The time zone from 122.73W-106.36W could be called "Western Time." On the Degree Confluence website, we use the 106th meridian to estimate 106.36W. Here is the list of states on this meridian:

Texas, New Mexico, Colorado, Wyoming, Montana, Saskatchewan (Canada), as well as Chihuahua, Durango, Sinaloa (Mexico)

Almost all of Texas is in Central Time (It's possible that 106.36W crosses through El Paso, though this can't be discerned from the Confluence website.) The other states are split. New Mexico, Colorado, and Wyoming are split more evenly -- the capitals Santa Fe, Denver, and Cheyenne are slightly on the Central side, but large parts of these states (especially Wyoming) are on the Western side of the line. Montana is almost completely in Western Time.

Sometimes the Continental Divide is suggested as a time zone line. This definition works well enough in the United States (and places more of Western New Mexico, Colorado, and Wyoming to be in the same time zone as their capitals). But it breaks down quickly outside the country, since the Divide actually crosses completely into other time zones.

Saskatchewan, at the northern end of this line, is a special case. It's considered to be on Year-Round Daylight Savings Time (which is what Florida is trying to accomplish). In its largest city, Saskatoon, the sun rises at 4:45 AM at the summer solstice and 9:15 AM at the winter solstice. Advocates of Year-Round DST might wish to check to see what Saskatchewan does about elementary schools that start before sunrise in the winter. (I did a quick check -- school start times there appear to be within 1/2 hour of winter sunrise.) Anyway, we might keep the entire province in Central Time on the Eleven Clock as well.

Because one of the time zones is based on the Greenwich meridian, noon and midnight are aligned for this time zone. Central Time, meanwhile, is the furthest away from alignment, since 90W is now a time zone border rather than the center longitude of its time zone. Those east of the meridian will observe noon on the Eleven Clock 33 minutes before -- and those west of the line will observe noon 33 minutes after -- noon Central Standard Time (Twelve-Hour Clock).

Here in California, noon on the Eleven Clock occurs about 22 minutes before noon Pacific Standard Time (Twelve-Hour Clock). Meanwhile, noon in the new 57.27W-40.91W time zone occurs about 14 minutes before noon Newfoundland Standard Time (Twelve-Hour Clock)

I've been assuming that Daylight Saving Time still exists on the Eleven Clock, with a one-hour biannual shift just as on the Twelve Clock. It's also possible to eliminate the biannual change completely for the Eleven Clock. One way to do this is to use a different reference meridian for each time zone, such as the easternmost longitude rather than the center longitude.

If we make this change, then noon in Greenwich occurs 33 minutes earlier than noon GMT, so it's nearly equivalent to Year-Round Half DST. The reference meridian for Central Time is now 90W, so noon is the same for Central Time on both clocks. Noon Western Time is one Eleven Clock hour after this, so it's about 5-6 minutes after noon Mountain Standard Time. Therefore noon in Arizona is always 5-6 minutes later on the new clock, and noon here is California is always 5-6 minutes than Pacific (Year-Round) Daylight Time. Noon in Saskatchewan is always the same on both clocks. So I prefer this as it's more easily convertible from the standard clock, at least where I live.

It's instructive to see what happens to the time zones in Australia, where Wendy Krieger lives. Well, almost all of the mainland lies in the 139.09E-155.45E and 122.73E-139.09E zones, but enough of Western Australia lies in the 106.36E-122.73E to suggest keeping three time zones rather than collapsing to two zones.

According to the Degree Confluence Project, The 122.73E border between Western and Central Time passes about midway through the state of Western Australia, near Gingerah in the north and Beaumont in the south. The 139.09E border between Central and Eastern Time is close enough to the existing border to justify not changing it at all.

If we follow the pattern above and use the easternmost reference meridian, then the new noon Western Time is 11 minutes earlier than the old noon, while the new noon Eastern Time is 22 minutes earlier than the old noon. The new moon Central Time, meanwhile, is 14 minutes later than the old noon rather than earlier. This is because Central Standard Time is 90 minutes ahead of Western Time, just as Newfoundland is 90 minutes ahead of New York. The new plan eliminates all of these half time zones that appear in both Canada and Australia.

New Zealand, meanwhile, is just like Greenwich, since it lies near the 180th meridian. The new noon is 33 minutes earlier than the old noon.

The new plan, if you recall, also eliminates Daylight Saving Time. Currently in Australia, the tropical states have Year-Round Standard Time (just like Hawaii in the U.S.). (And of course, as I wrote back in my November 6th post, DST in Australia currently follows the seasons in that hemisphere, so in March they'll set the clocks back an hour and spring forward in October.)

It's also possible to have a worldwide Sheila Danzig plan, where there are only 11 time zones, each two hours apart. We'll combine the time zones listed above:

171.81W - 139.09W 2 AM
139.09W - 106.36W 4 AM
106.36W - 73.64W 6 AM
73.64W - 40.91W 8 AM
40.91W - 8.18W 10 AM
8.18W - 24.55E 1 PM
24.55E - 57.27E 3 PM
57.27E - 90.00E 5 PM
90.00E - 122.73E 7 PM
122.73E - 155.45E 9 PM
155.45E - 171.81W midnight

The Danzig plan often results in late winter sunrises in the north, so I'd recommend returning to 180 as a reference meridian rather than 171.81W.

The mainland U.S. is divided into two time zones just as on Danzig's original plan, but the border between them is farther west. Australia also has only two time zones, with only Western Australia in a different zone from the rest of the country. New Zealand keeps its clock, but Greenwich Time no longer exists, as its time zone is on the equivalent of Year-Round DST.

Before we leave the Eleven Calendar, I want to take one last look at the 11-day week. I've written before what this long week looks like. Since at least half the week needs to be workdays, there must be six workdays. But as six straight days of work are exhausting, we have a midweek day off. So the pattern is three days of work, one day of rest, three days of work, and then a four-day weekend.

The six-day workweek works well with high school block schedules. For example, the high school I subbed at last week (in my old district) has a block schedule. Tuesdays and Thursdays are even days, while Wednesdays and Fridays are odd days. Mondays are all-classes days. Of course, the only reason for the all-classes days is the number of school days in a week, five, is odd. With six-day weeks, there's no need for an all-classes day on Monday. We can have a pure block schedule, with three even days and three odd days every week.

This year, I've subbed at middle schools in my new district. These middle schools use a rotating schedule, where the first day is periods 1-2-3-4-5-6, then the second day is 2-3-4-5-6-1, the third day 3-4-5-6-1-2, and so on. There is no relationship between the day of the week and which period starts the day -- at least, there isn't with a five-day school week. A six-day school week, of course, fits the middle school rotation schedule like a glove. The three days before the midweek break start with periods 1, 2, and 3, and the three days after the break start with 4, 5, and 6. This is another argument in favor of the six-day week on the Eleven Calendar.

Wow, I know -- my posts on calendars, clocks, and DST always go long. I still haven't said much about today's lesson on the Fundamental Theorem of Similarity. Well, I'll only keep some of what I wrote last year since this post is so lengthy.

This is what I wrote last year about today's lesson:

Section 12-6 of the U of Chicago text covers the Fundamental Theorem of Similarity. As its name implies, it is the most important theorem related to dilations and similarity. Here is how this theorem is stated in the U of Chicago:

Fundamental Theorem of Similarity (U of Chicago):
If G ~ G' and k is the ratio of similitude, then
(a) Perimeter(G') = k * Perimeter(G) or ...
(b) Area(G') = k^2 * Area(G) or ...
(c) Volume(G') = k^3 * Volume(G) or ...

Notice how I had to rewrite this theorem so that it fits into ASCII. Here the * and ^ symbols denote multiplication and exponentiation, respectively -- these symbols should be recognizable as they appear on TI graphing calculators. The "or ..." sections refer to the text rewriting each equation as a ratio, so that the ratio of the perimeters is k, the ratio of the areas is k^2, and so on, but that is rather awkward to write in ASCII.

David Joyce describes this theorem in his criticism of the Prentice-Hall text:

The proofs are omitted for the theorems which say similar plane figures have areas in duplicate ratios, and similar solid figures have areas in duplicate ratios and volumes in triplicate rations. At least there should be a proof that similar triangles have areas in duplicate ratios; that's easy since the areas of triangles are already known.

But we won't be proving the U of Chicago or Prentice-Hall versions of this theorem. Instead, we will be proving Dr. Wu's version of the Fundamental Theorem of Similarity... least we would be if I haven't spent so much time writing about, well, time. And after all, we're following only the U of Chicago text this year, not Dr. Wu.

Here is the U of Chicago proof:

(a) Perimeter is just the sum of the lengths. Suppose lengths a, b, c, d, e, ...make up the perimeter of G. Then lengths ka, kb, kc, kd, ke, ...make up the perimeter of G'.

Perimeter(G') = ka + kb + kc + kd + ke + ...
                       = k(a + b + c + d + e + ...)
                       = k * Perimeter(G)

(b) Let A = the area of G. Then you could think of the area of G as the sum of areas of A unit squares. Then the area of G' is the sum of areas of A squares k units on a side. Since each square in G' has area k^2,

Area of G' = A * k^2 = k^2 * Area of G.

(c) The argument is identical (except with unit cubes). QED

As usual, I keep the old worksheet, so here's the Wu proof, in case you need it for the worksheet:

Given: OP' = 3OPOQ' = 3OQ
Prove: P'Q' | | PQ, P'Q' = 3PQ

Statements                                    Reasons
1. OP' = 3OPOQ' = 3OQ           1. Given
2. W on ray PQ with PW = 3PQ,  2. Ruler/Point-Line Postulate
    on ray QQ' with OQ = QU
    V on ray QW with PQ = QV
3. UV | | Q'WQ'W = 2UV             3. Definition of Midpoint and Midpoint Connector Theorem
4. QQ' = 2OQQW = 2PQ            4. Betweenness/Segment Addition Theorem
5. OPUV is a parallelogram          5. Diagonals Bisect Pgram Test
6. UV | | OPUV = OP                  6. Pgram Consequences
7. OP (same as P'P) | | Q'W           7. Transitivity of Parallels Theorem
8. PP' = Q'W                                  8. Algebra and Substitution
9. PP'Q'W is a parallelogram        9. One Pair Parallel & Congruent Pgram Test
10. P'Q' | | PWP'Q' = PW            10. Pgram Consequences
11. P'Q' | | PQ, P'Q' = 3PQ           11. Substitution

That was what I wrote last year about today's topic. In the past, I didn't cover Lesson 12-6 per se, but I did include an extra activity page that incorporates Lessons 12-6 and 12-7, so it's logical to include it here.


  1. A very common device that i use to work with numbers that do not divide 360, is to pre-introduce the desired factor into the number.

    So if you are discussing 22 time zones, multiply the angle by 1.1, so Brisbane at 150 E becomes 165 E.

    On the eleven-calendar, it is interesting to observe that if the egyptians used an 11-hour daylight, rather than a 10-hour one, and an 11-day 'decan' (a 10-day week that functions as a zodiac sign), you would have rising at night 11 decans, and thus still have a 24-hour day. But you can get it back to 22 by having nine divisions on the sundial and 2 off it (twilights).

    The sundial is good for in the order of 100 or so divisions. You see the chinese cé (variously 1/96 or 1/100 day), and the OE moment (1/4 hour) does this. So you can easily get something like 242 divisions (22 hours of 11 6-minute "uncia".

  2. Hmm, this is interesting. So now there are 360 * 1.1 = 396 degrees in a full circle. This reminds me of 400 "gradians" in a circle, since 396 is so close to 400.

    My own longitude is extremely close to 130W under this system, so my time would be seven hours behind Greenwich, while your time would be nine hours ahead of Greenwich.

    Thanks for an interesting sundial idea as well!