The Sierpinski Cake

June 14, 2018 by

The verdict is in:  despite my less-than-perfect cutting and decorating skills, a Sierpinski cake does indeed look pretty cool.

Sierpinski Triangle made of cake

I had been so distracted by the descriptions of triangular cupcake liners that I didn’t order any, but certainly those would have made for more equally shaped triangles and a crisper looking design.  This was made by baking a 9″×13″ cake, cutting it into 3 long strips, and doing a zigzag pattern on each to get 9 equilateralish triangles plus a bit extra.  Now I just need an extra-large platter for serving, because we are SO having this again sometime…

 

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The mysterious triangle

June 12, 2018 by

I was just looking up triangular cupcake molds, as one does — or at least, as one does after the upcoming birthday child asks if their cake can be made into a Sierpinski triangle, and you both mull about it and then realize that that would be CRAZY because then no one would get any cake, and then you get distracted wondering if it could be approximated with cupcakes and you realize that would be awesome and the next time you make cupcakes they really should be displayed in a Sierpinski triangle — and  upon looking at a couple pages of molds I noticed a funny thing.

Amazon has trouble with triangles.

Granted, I only saw oddities on two pages, but since that’s 100% of the pages I looked at, it was still a pretty high percentage.

Exhibit #1:  Triangle is now a color:

Color:Triangle

Exhibit #2:  Triangles have a diameter:

AmazonTriangleDiameter

Of course, triangles do in fact have a diameter, since there’s a maximal distance between points (which turns out to be the length of the longest side).  So presumably these are 2″ per side, and Amazon gets bonus points for an unnecessary use of Diameter.  I admit, though, that part of me thinks that what they called “diameter” was really the distance from a vertex to the opposite side, because from a visual perspective I think that seems the most like the diameter of a circle.  That would mean that the actual side length was (2/√3)*2″, or about 2.3″.  The other cupcake liners were 2″ to a side at the base and 2.4″ to a side at the top, which means that either number would be consistent even if these were the same size.

So in the end, I’m not sure of the answer.  What I am certain of, however, is that the Sierpinski approximation cupcake is an awesome idea, and should become a reality as soon as possible.

 

The rate of change of gasoline

May 4, 2018 by

Is it better to fill up a gas take once a week for $80, or put in a quarter tank 4 times a week for $20 each time?  That question does have two reasonable answers, depending in no small part on whether you have access to $80 or just to $20 at a time, but what isn’t in doubt is that four quarter-fill-ups at $20 each isn’t actually cheaper overall than one full-fill-up at $80.  Or at least, that shouldn’t be in doubt.

There’s an article about it here, but it doesn’t lessen the confusion at all.

(The reference to the question of  which is closer, the West Cost or the Moon, is a reference to a discussion from a year ago.)

Math Mistake (sort of) – the problem with negatives

May 2, 2018 by

It has been a while since we’ve seen a math mistake in the news, but a recent search turned up an old one that I’d never seen (Thanks TwoPi for pointing it out!)  And the funny thing is, it’s not actually a mistake at all – the math is correct.  And that’s the problem.

Back in November of 2007, the National Lottery in the United Kingdom had a new scratchoff ticket for their “Cool Cash” Lotto.  The idea behind the game was the a person would scratch to reveal a specific temperature — say, 15º — and would then scratch to reveal  three more temperatures.  If any of these three numbers was lower than the Chosen Special one (15º in this example), then the person won a prize.  Hooray!

But this was in the UK, which uses Celsius, and negative temperatures are pretty common in the winter.  So the target temperature might be something like –7º, and the three additional temperatures might be –6º, –5º, and –4º.  From a mathematically correct point of view,  that’s not a winning ticket because all the numbers are above –7º.  But people who focused on the numberals 6, 5, and 4, all of which are less than 7, thought they’d won.

It took but a day for this to become a problem, and after no small amount of confusion on the part of customers and shopkeepers, the tickets were pulled.  They had lasted less than a week.  Lottery we hardly knew ye.

For more details, including a video, see the article in the Manchester Evening News:
https://www.manchestereveningnews.co.uk/news/greater-manchester-news/cool-cash-card-confusion-1009701

Calculating Fahrenheit

August 15, 2016 by

Thermometer_Hotel_BaronThe post about the math mistake in temperature conversation reminded me of a formula that a friend told me about (thanks DSD!).  She was traveling abroad, and the guide she was with said that to convert Celsius to Fahrenheit people should use the formula:

Double the Celsius, and subtract from it the amount obtained by moving the decimal place one unit to the left.  Then add 32 to get the corresponding Fahrenheit.

For example, with a temperature like 50°C, you’d double 50 to get 100, then from that subtract 10.0 to get 90.  Finally, you’d add 32 to 90 to get 122°F.

This is equivalent to the formula
Temp in °F = (9/5) (Temp in °C) + 32.

In particular, if C is the temperature in Celsius, the description to double and the subtract that amount with the decimal place moved describes 2C – 0.1(2C), which is 1.8C, or 9/5C.

It does seem to me to be quicker to compute 9/5C by doubling C and subtracting a tenth of the result than to multiple by 9 and divide by 5 in some order.  The conversion isn’t as quick as “Double and add 30″*, perhaps, but unlike that estimation it has the advantage of being exact.

*a formula that always brings to mind the movie Strange Brew

The thermometer is by Bernard Gagnon – Own work, CC BY-SA 3.0. It has Centigrade rather than Celsius at the top, which I found interesting since I remember learning both terms in school.

Another Math Mistake:

August 13, 2016 by

640px-OmnogoviLandscape (1)

This mistake was printed almost a year ago, but it’s still relevant, and math mistakes never go out of style.  This was posted by Richard Fuhr, who I believe is the original author.

The author was looking at an article about the Gobi desert in China, which read in part: “Temperatures may vary up to 95°F (35°C) in one day in the Gobi.”  It also indicated that the average temperature in winter was -40°F (-40°C) and in the summer could be 122°F (50°C)

The -40°F being equal to -40°C is correct – it’s the only place the two temps have equal numerical designation, and I am a little sad that I’ve never gotten to experience it except in windchill form.  The 122°F being equal to 50°C is also correct, and something I have exactly no desire to experience, although it’s still lower than the 129.2°F (54°C) recorded in Kuwait last month.  Both of those conversations can be found by using one of the formulas

  • Temp in °C = (5/9) (Temp in °F – 32)
  • Temp in °F = (9/5) (Temp in °C) + 32.

The issue is that these are temperature readings, not changes in temperature.  For a change in temperature, the 32 in either formula will disappear, leaving

  • Δ°C = (5/9) (Δ°F )
  • Δ°F = (9/5) (Δ°C)

This means that a variation of temperature of 95°F would actually correspond to a change of about 52.8°C, not 35°C.  And a variation of 35°C would be a change of “only” 63°F, not 95°F.    It’s not possible to tell mathematically whether the correct variation was  95°F (53°C) or 63°F (35°C), but looking through The Internet at temperature variations, it appears to me that although either one is possible, the printed variation was likely intended to be 35°C, not  95°F.

The photo above is by Doron, with a Creative Commons license.  Thanks to YG for bringing the original article to my attention!

 

Archimedes Spiral on the Lunar Module

July 30, 2016 by

apollo_11_lunar_module_eagle_in_landing_configuration_in_lunar_orbit_from_the_command_and_service_module_columbiaOne of our alums (Thanks CJ!) sent a link this summer aarchimedean_spiralbout how MATH was used in the design of the Alignment Optical Telescope in the Apollo Lunar Modules.  I mean, yes, of course it was, but in particular an Archimedes Spiral — a spiral where the distance is increasing steadily, such as r = θ in polar coordinates — was used instead of a heavier piece of equipment.  This is, I think, the first time I’ve seen a modern application of the Archimedes spiral.

The video is available below, and there is more info  at NASA.

Have a power-ful day

February 4, 2016 by

balloonsToday is 2/4/16 or 4/2/16, depending on where you live and how you write dates.  Either way, it’s a great day because 24=16 and 42=16.  There aren’t many days like that (although we are treated to two this year), so it’s worth taking a moment to celebrate.

Happy Pi Day 2015!

March 10, 2015 by

Pi Day is coming up, and this year is a special one:  instead of just 3.14 on March 14 we get 3.1415.  Woo hoo!  So grab a Sudoku

(from Brainfreeze Puzzles:  Digits 1-9 in each row, column, and square, plus digits 31415926 in each block of pink)

and grab a beverage of your choice

GodzillaPiBeer2015

and enjoy.  Happy Pi Day!

How many grams in an ounce?

March 9, 2015 by

Converting between units can be hard, as seen before (and before and before).  Fortunately, food containers often include both English units and Metric units.  Unfortunately, those two don’t always match.  Take, for example, Producers Sour Cream.  Their 32 oz container says it has 907 grams, which is about what you’d expect.  The 16-ounce container has half has many.  Not half of 907, but half of that again:  in bold defiance of the laws of physics, it sports a mere 226 grams.

Producers Sour Cream

This mistake has apparently gone on for years.  What’s equally strange is that the various nutrition sites that include information about this product also say 16 oz (226g) without comment.     Because, as stated above, units are hard.

Thanks to Philip Bailey for bringing this to our attention!  And speaking of Math Mistakes, as I was, several of the mistakes listed in this very blog are published in the PRIMUS article “Math Mistakes that Make the News” by Yours Truly, which can be downloaded for free during the month of March (2015).

 

Math Mistake…tell your (two) friends

September 17, 2014 by

Back in the 80s, there was a commercial for Faberge Organic Shampoo.  And even if the shampoo doesn’t sound familiar, you might have heard of the ad (“…and they tell two friends…”)

Hey, it’s exponential functions!  1 friend tells 2 friends, those 2 friends tell 4 friends, those 4 friends tell 9 friends, those…wait, 9?  Where did that come from?  And then those 9(?) friends tell 16 people.  So it almost works, except that after the photo of 2 people they decided to switch to perfect squares.

Fortunately, a later ad brings the whole thing to a halt before reaching 9:

Good job Faberge people – you skipped the 9!  Of course, this one went straight from 1, 2, 4 to 16 before diving headlong into a grid of 24 people, so I’m not sure it was much of a mathematical improvement.

Threesixty360…your source for commenting on 30 year old math mistakes that have already been well documented.

Long lines on Earth

September 15, 2014 by

In spherical geometry, the shortest-length curve between two points on the surface of the sphere turns out to be part of a Great Circle – an equator-line circle that cuts the sphere in half.  So lines are circles, which is fun to share with philosophers.  (Note – taxicab geometry provides that same amusement, where circles are squares.)

So a natural question, where “natural” means I never actually thought of it but wish I had, is What is the longest line along the surface of the earth that goes entirely through water?  This would be the longest possible straight-line sailing distance, if you ignored all the physical aspects of sailing like wind and water currents.  Fortunately, before I even thought of the question, someone had answered it.  Behold!

Longest straight-line distance through the water on earth.

This gif appears to be from a youtube video by Patrick Anderson of 2012 (here) which has the advantage of being a little slower.

So that raises the question of the longest straight-line distance through land.  And here’s a guess at it:  http://i.imgur.com/nbNfl.jpg and then another one  https://sites.google.com/site/guybruneau/fun-stuff/longest-distance-on-land, although that second one it doesn’t quite look like part of a Great Circle so possibly the projection imposed a different geometry. Or possibly I have trouble visualizing projections of Great Circles, which is also possible because they are weird.  (The cool kind of weird, of course.)

Thanks CJ for sending me that gif, although now that I’m finding myself asking questions like “What line passes through the most countries?” I can tell that it’s going to keep me from my grading for longer than it should.

Decimal Mistake in the News

July 23, 2014 by

Pennies. Not all from the US, however.Decimal points are small, and so easy to lose.  And it appears that many of them were lost on FAFSA (Free Application for Federal Student Aid) forms, which is NOT a place that you would want incorrect data.  According to an official document from July 18, people filling out the form were supposed to round monetary values to the nearest dollar, rather than using exact dollar-and-cents amounts.  But some people put down cents anyway, and the computer didn’t alert them, or tell them there was a problem.  No, it slyly accepted the amounts, and then threw all the decimal points in the trash, so [as the official memo said], and income that had been recorded as $5000.19 was suddenly interpreted as $500019, which is one heck of a sweet income and probably enough to disqualify you from most financial aid.

This didn’t happen with just a couple people, either – The Wire says that 200,000 people are likely to be affected.  And because it’s more than just a couple, schools have to look at all those applications, every single one, to catch any errors.  Those errors might be that people didn’t get aid who should, which is a bummer, but it could also mean that people got too much money.  That doesn’t sound as bad initially, but the July 18 memo says, ” If such aid has already been disbursed the institution may need to change awards and return (or have the student return) any overawarded funds.”  I can’t imagine that it will go over all that well for a school to tell someone to give back money that was promised, so I suspect this messiness will last a while.

Hat tip to Yousuf for pointing out this article!

What is an urn?

June 25, 2014 by

Maya_funerary_urnI can think of exactly two times when the word urn is used: as a container for someone’s ashes, and as a container for colored balls.  Since I’ve never physically seen an urn that has balls in it, it makes me wonder – when did that become such a standard in probability problems?  Why are the balls in an urn in the first place?

When I asked that very question, TwoPi mentioned “surmounted” as another example of an English word that seems to be used exclusively in one context:  to describe Norman Windows (a window in the shape of a rectangle surmounted by a semicircle).   Notre-Dame_Etretat_nef3“surmountable” is more common, and “insurmountable” even more so, so I suppose “surmounted” does actually appear in a related context (as in “that difficulty has been surmounted”), but it’s still relatively unusual.  I suspect that there are other words, English words as opposed to mathematical terms, that just don’t show up very often outside of the exercises in a text.

Here is a Norman Window, by the way, from Notre-Dame d’Étretat in Étretat, France.

Gene sharing math

June 22, 2014 by

ConsanguinitetThe other day I found myself wondering what proportion of genes cousins would expect to share compared to biological siblings.  This took more time to figure out than I would have expected, in part because I knew that siblings who share two parents have 1/2 their genes in common on average, so I thought cousins sharing two grandparents might have a quarter.  They don’t, though – it’s half that.  In reasoning it out, it turned out to be easiest to think of moving up the biological family tree to a common ancestor, which led to one general formula and a few specific cases:

The Generalization:

Given two people A and B, find their closest common ancestor C.  If there are n generations from A to C and m generations from B to C, then the expected proportion of shared genes is (½)n+m.  If there are two closest common ancestors (for example, both parents) then this number would double.

In the case of a parent and child, for example, there is 1 generation from the child to the parent (the common ancestor) and 0 from the parent to itself, so the proportion of shared genes would be (½)1, or just ½.  Cousins would each be 2 generations from common grandparents, leading to (½)4, or 1/16, for cousins with one grandparent in common (sometimes called half cousins) and twice that for cousins with two grandparents in common (sometimes called full cousins).  Double cousins  — that is, people who are cousins on both sides of the family tree (for example, cousins whose mothers are sisters and whose fathers are brothers) — would still have grandparents as the closest common ancestor, but now it would be up to four common grandparents instead of just one or two: the expected proportion of shared genes between cousins with four common grandparents would be 4·(½)4, or just ¼.  Likewise, an aunt and nephew with two parents/grandparents in common would be 1 and 2 generations respectively from this pair of common ancestors, so the expected proportion of shared genes would be 2·(½)3, also ¼.

Special Case 1:  great-great-…-great grandparents
In this case the older relative is the common ancestor, so if “g” is the number of “great”s then the proportion of shared genes is (½)g+2.  The additional 2 in the exponent is because the number of “great”s counts the generations after grandparents, who are already 2 generations away from their grandchildren.  This is the only case where the proportion is exact:  in all the others, it’s only an expected proportion because siblings could have anywhere from no overlap of genes to complete overlap of genes from each common parent.

Special Case 2:  great-great-…-great aunts and uncles
In this case the older relative’s parent(s) are the common ancestor.  With a great-uncle and great-niece, for example, the great-uncle’s parent(s) are the great-grandparent(s) of the great-nephew.  This means that there is 1 generation from the great-uncle to his parent(s), but 3 from the great-niece to that common ancestor, with each additional “great” adding another generation.  If “g” is the number of “great”s, then the expected proportion of shared genes would be (½)g+3 if there is one parent in common, and (½)g+2 if there are two.  (I personally find it interesting that you can expect to share the same proportion of genes with a sibling who shares both parents as you do with either of the individual parents, the same proportion with an aunt or uncle who shares both grandparents as you do with either of the individual grandparents, and the same proportion with a great-great-…-great aunt/uncle who shares both great-great-…-great grandparents as you do with either of those great-great-…-great grandparents themselves.)

One clarification:  great-aunt is the term I grew up with, but in looking around I just discovered that “grand-aunt” may be the technically correct term, since that person is in the same generation as a grandparent; likewise, the sister of a great-grandparent would be a great-grand-aunt.  This appeals to me aesthetically.   If you were to use these terms, then you’d have one fewer “great” in describing the relationship, and you’d need to add 1 to the exponent in the formulas above.

Special Case 3:  second cousins once removed (and the like)
Cousins share at least one grandparent, second cousins share at least one great-grandparent, and xth cousins share at least one great(x-1) grandparents.  This means that xth cousins are each (x+1) generation removed from the common ancestor(s), and would expect to share (½)2x+2 of their genes if there is one common relative and (½)2x+1 if there are two. Each removal  refers to one of the people being one more generation removed from any common ancestors, and so increases the power of ½ by 1.  This means that xth cousins who are y-times removed would expect to share (½)2x+y+2 of their genes if there is one common relative and (½)2x+y+1 if there are two. Second cousins once removed would share either (½)7 or (½)6 of their genes, while first cousins twice removed would share (½)6 or (½)5.

For those who like the visual, there is a handy little chart below, which appears to be in the public domain on Wikipedia. It does make some assumptions, however – namely, that siblings, cousins, aunts and nieces, etc. have exactly two closest relatives in common (both parents, two grandparents, etc.).

Cousin_tree_(with_genetic_kinship)