"What distinguishes mathematical knots from the everyday knots one ties, is that they have no ends. They are a closed type of loop, which cannot be formed into a circle."
We've returned to the section on knots. Here Pappas provides the mathematical definition of a knot as a closed loop that can't be deformed to a circle (the "unknot"). Therefore the ring structures from Serra are not knots, since they aren't a single closed loop. The Olympic rings are five circles, not one.
Pappas tells us that knots can't exist in more than three dimensions. The reason is that all closed loops in 4D can be deformed to the unknot. Instead, we can knot a plane in 4D. One example of a knotted plane is the Cartesian product of any 3D knot and any line segment.
According to Pappas, the simplest knot is the trefoil knot. Here is a Wolfram link to this knot:
http://mathworld.wolfram.com/TrefoilKnot.html
The "tre-" in trefoil means three, and indeed this knot has three crossings ("crossing number 3"). She tells us that the trefoil is the only knot with crossing number 3. The trefoil is closely related to its reflection image. Even though there is an isometry mapping one to the other, no isotopy (continuous deformation) exists.
Pappas tells us that there is only one knot of crossing number 4 (the "figure-eight knot") and two knots of crossing number 5. But the number of knots grows rapidly -- she states that there are over 12,000 knots of crossing number at most 13.
As usual, there is a sequence on OEIS -- the number of prime knots with n crossings. (I mentioned the OEIS, the online sequences website, on the second day of school last year.)
https://oeis.org/A002863
The final example Pappas gives on this page is the composite of a knot and its reflection. She tells us that to our surprise, this cannot deformed to the unknot. She writes that if we try to deform it:
"They simply pass through one another and remain unchanged."
There's not much on this Pappas page that is teachable in a middle school science class at all. So instead, let's go straight into Serra.
Lesson 0.8 of Serra's Discovering Geometry is called "Perspective." This is the second of two sections appearing in the Second Edition but not in the modern editions. Serra begins:
"Many of the paintings created by European artists during the Middle Ages were commissioned by the Roman Catholic Church. The art was symbolic; that is, people and objects in the paintings were symbols representing religious ideas."
Unlike Lesson 0.5 on mandalas, which we choose to include on the blog even though it's "missing" from the modern editions, Lesson 0.8 can just be left out altogether. This is because we'll be starting the U of Chicago text next week, and that text already has a lesson on perspective (Lesson 1-5), so Day 15 would just be a repeat of Day 8.
Then again, we recall that in my class last year, the students in all grades had trouble drawing cubes even though those were on isometric paper rather than in true perspective). So we might wish to teach perspective on both Day 8 and Day 15. True perspective drawings should most likely be completed on plain unlined white paper, with a straightedge to draw lines toward the vanishing point. Lined notebook paper for one-point perspective drawing may also be acceptable -- but not for two-point perspective (the subject of today's worksheet).
The worksheet below comes from "marcandersonarts" and "Daisuke Motogi."
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