Math Maps The Island of Utopia

Under the category of “Somebody Had To Figure It Out,” Dr. Andrew Simoson of King University (Bristol, Tennessee) used calculus to determine the shape of the island of Utopia in the 500-year-old book by Sir Thomas More based on the description of island given in the book’s introduction.

News article: https://www.insidescience.org/news/math-maps-island-thomas-mores-utopia

Paper by Dr. Simoson: http://archive.bridgesmathart.org/2016/bridges2016-65.html

This Is Why There Are So Many Ties In Swimming

From the excellent article “This Is Why There Are So Many Ties In Swimming“, ties in swimming are allowed by the sport’s governing body because of the inevitability of roundoff error.

In 1972, Sweden’s Gunnar Larsson beat American Tim McKee in the 400m individual medley by 0.002 seconds. That finish led the governing body to eliminate timing by a significant digit. But why?

In a 50 meter Olympic pool, at the current men’s world record 50m pace, a thousandth-of-a-second constitutes 2.39 millimeters of travel. FINA pool dimension regulations allow a tolerance of 3 centimeters in each lane, more than ten times that amount. Could you time swimmers to a thousandth-of-a-second? Sure, but you couldn’t guarantee the winning swimmer didn’t have a thousandth-of-a-second-shorter course to swim. (Attempting to construct a concrete pool to any tighter a tolerance is nearly impossible; the effective length of a pool can change depending on the ambient temperature, the water temperature, and even whether or not there are people in the pool itself.)

Pokemon Go and the Course Syllabus

This is definitely going on my course syllabus this semester:

Source: https://www.facebook.com/PokeGoMemePage/photos/a.249688652081224.1073741827.249683562081733/260297621020327/?type=3&theater

Predicate Logic and Popular Culture (Part 92): Annie Get Your Gun

Let $W(x,y)$ measure how well $x$ can do $y$ . Translate the logical statement

$\forall y (W(\hbox{you},y) < W(\hbox{I},y))$ .

This needs no further introduction:

Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 91): The Scarlet Pimpernel

Let $H(x,t)$ be the proposition “ $x$ happens at time $t$ ,” let $V(x)$ be the proposition “ $x$ is a valley,” let $M(x)$ be the proposition “ $x$ is a mountain,” let $S(x,y,t)$ be the proposition “ $y$ must scale $x$ at time $t$ ,” let $W(x)$ be the proposition “ $x$ are perilous waters,” and let $S(x,y,t)$ be the proposition “ $y$ must sail $x$ at time $t$ .” Translate the logical statement

$\forall t \exists x (V(x) \land H(x,t)) \land \forall t \exists x(M(x) \land H(x,t) \land \exists y S(x,y,t))$

$\land \forall t \exists x(W(x) \land H(x,t) \land \exists y S(x,y,t))$ .

This matches the second half of the opening verse of the showstopper “Into The Fire” from the musical The Scarlet Pimpernel.

I also recommend Steve Amerson’s rendition:

Predicate Logic and Popular Culture (Part 90): The Beatles

Let $N(x)$ be the proposition “You need $latex x.” Translate the logical statement

$N(\hbox{love}) \land \forall x (x \ne \hbox{love} \Rightarrow \lnot N(x))$ .

This matches the title of one of the Beatles’ greatest hits.

Predicate Logic and Popular Culture (Part 89): Into the Woods

Let $S(x)$ be the proposition “ $x$ is on your side” and let $A(x)$ be the proposition “ $x$ is alone.” Translate the logical statement

$(\exists x S(x)) \land (\forall x \lnot A(x))$ .

Of course, this matches the last two lines of “No One Is Alone,” which was released as a movie in 2014.

I’m personally partial to Steve Amerson’s rendition of this song:

Predicate Logic and Popular Culture (Part 88): Bob Dylan

Let $H(x)$ be the proposition “You have $x$ ,” let $L(x)$ be the proposition “You have $x$ to lose,” let $p$ be the proposition “You’re invisible now,” and let $S(x)$ be the proposition “ $x$ is a secret to conceal.” Translate the logical statement

$((\forall x \lnot H(x)) \Rightarrow (\forall x \lnot L(x))) \land p \land \forall x (S(x) \Rightarrow \lnot H(x))$ .

This matches the last two lines of the closing verse of this classic from Bob Dylan.

Predicate Logic and Popular Culture (Part 87): Bruce Springsteen

Let $R(x)$ be the proposition “ $x$ needs a place to rest,” let $H(x)$ be the proposition “ $x$ wants to have a home,” and let $A(x)$ be the proposition “ $x$ wants to be alone.” Translate the logical statement

$\forall x (R(x) \land H(x) \land \lnot A(x))$ .

This matches three of the lines in the closing verse of one Bruce Springsteen’s great hits from the 1980s.

Predicate Logic and Popular Culture (Part 86): My Fair Lady

Let $T(t)$ be the proposition “ $t$ is at night” and let $D(t)$ be the proposition “I could have danced at time $t$ .” Translate the logical statement

$\forall t (T(t) \Rightarrow D(t))$ .