"Over the ages the triangle, the square and the rectangle have played major roles in architectural design. Wood and stone were among the first natural materials builders used to make their shelters."
This is the first page of a new section, "21st Century Architecture -- Space Filling Solids." But Pappas begins by describing buildings from well before the 21st century (unless she means B.C.).
Pappas tells us that when builders used simple materials, the structures had simple shapes. This is why she connects wood and stone buildings to triangles (think pyramids), squares, rectangles, and eventually arches and domes (in Roman and Gothic times). And as new building materials were developed (as in steel, iron, glass, concrete, and bricks), new shapes were used in the design. So just as building materials became more sophisticated, so did the underlying geometry.
Indeed, Pappas writes:
"Later, plastics and synthetic materials, along with tensegritory structures, allowed architects to consider a whole new family of shapes."
Notice that "tensegritory" actually means "tensional integrity." It was first used by Buckminister Fuller and defined by Pappas on page 252 -- one of the pages we skipped due to the weekend.
There are two drawings on the current page 254. Here is the first caption:
"A diagram from Leonardo da Vinci's notebook, showing his work on trying to resolve the forces acting on the arch."
And here is the second caption:
"Gothic plans by the master architect of the Dome of Milan, Caesar Caesariano."
All of this reminds me of the Classical Curriculum, where literature, history, and science lessons are taught in sync. I wonder whether it's possible to tie in geometry to this idea as well. So the first year, when ancient history is taught, students learn about triangles, squares, and rectangles. Then in the second year, they learn about medieval history and thus more complex shapes that appeared in buildings of the time, and so on.
Well, it's just a thought. Of course, we have to deal with the fact that there is more to mathematics than just geometry and the shapes of structures.
All of this is about two-dimensional blueprints of three-dimensional buildings. Let's drop it down a dimension, though, and think about just one dimension for today's lesson.
Lesson 1-8 of the U of Chicago text is called "One-Dimensional Figures." (It appears as Lesson 1-6 in the modern edition of the text.)
This is what I wrote two years ago about today's lesson:
Lesson 1-8 of the U of Chicago text deals with segments and rays. The text begins by introducing the simple idea of betweenness. In Common Core Geometry, betweenness is an important concept, because it's one of the four properties preserved by isometries (the "B" of "A-B-C-D").
As I mentioned a few days ago, for Hilbert, betweenness is a primitive notion -- an undefined term, just as point, line, and plane are undefined. Yet the U of Chicago goes on to define it! It begins by defining betweenness for real numbers:
"A number is between two others if it is greater than one of them and less than the other."
Then the text can define betweenness for points:
"A point is between two other points on the same line if its coordinate is between their coordinates."
But Hilbert couldn't do this, because his points don't have coordinates. Recall that it was Birkhoff, not Hilbert, who came up with the Ruler Postulate assigning real numbers to points. Instead, Hilbert's axioms contain statements about order (Axioms II.1 through II.4), such as:
"II.2. If A and C are two points of a line, then there exists at least one point B lying between A and C."
Since we have a Ruler Postulate (part of the Point-Line-Plane Postulate), this statement is obvious, since points have coordinates and the same is true for real numbers -- between reals a and c is another real b.
I've seen some modern geometry texts mention a Ruler Postulate, but nonetheless leave the term betweenness undefined. Now as we mentioned earlier with point, line, and plane, if a term such as betweenness is undefined, then we need a postulate to describe what betweenness is. This postulate is often called the Segment Addition Postulate:
"If B is between A and C, then AB + BC = AC."
Notice that this statement does appear in the U of Chicago text. But the text doesn't call it the Segment Addition Postulate, but rather the Betweenness Theorem. As a theorem, we should be able to prove it -- and since after all, the text defines betweenness in terms of real numbers, we should be able to use real numbers to prove the theorem. Indeed, the text states that we can use algebra to prove the theorem, but the proof is omitted.
Following David Joyce's admonition that we avoid stating a theorem without giving its proof, let's attempt a proof of the Betweenness Theorem. We are given that B is between A and C. Now let us assign coordinates to these points. To make it easy to remember, we simply use lowercase letters, so point A has coordinate a, point B has coordinate b, and point C has coordinate c.
We are given that B is between A and C, so by definition of betweenness, we have either a < b < c, or the reverse of this, a > b > c. Without loss of generality, let us assume a < b < c (especially since the example in the book has a < b < c). Now by the Ruler Postulate (the Distance Assumption in the Point-Line-Plane Postulate), the distance between A and B (in other words, AB) is |a - b|. Since a < b, a - b must be negative, and so its absolute value is its opposite b - a. (To avoid confusing students, we emphasize that to find AB, we just subtract the right coordinate minus the left coordinate, so that AB isn't negative. This helps us to avoid mentioning absolute value.) Similarly BC = c - b and AC =c - a. And so we calculate:
AB + BC = (b - a) + (c - b) (Substitution Property of Equality)
= c - a (simplification -- cancelling terms b and -b)
= AC
The case where a > b > c is similar, except that AB is now a - b rather than b - a. All the signs are reversed and the same result AB + BC = AC appears. QED
Don't forget that I want to avoid torturing geometry students with algebra. And so I simply give the example with numerical values, with the variables off to the side for those who wish to see the proof.
The text proceeds to define segments, rays, and opposite rays in terms of betweenness. Notice that these definition are somewhat more formal than those given in other texts. A typical text, for example, might define a segment as "a portion of a line from one endpoint to another." But the U of Chicago text writes:
"The segment (or line segment) with endpoints A and B is the set consisting of the distinct points A and B and all points between A and B."
The definitions of ray and opposite ray are similarly defined in terms of betweenness.
The section concludes with the notation for line AB, ray AB, segment AB, and distance AB. But although every textbook distinguishes between segment AB and distance AB, many students -- and admittedly, many teachers as well -- do not. The former has an overline, but the latter doesn't. Unfortunately, Blogger allows me to underline AB and strikethrough
Now if
To avoid confusion, in the following images I threw out Question 8 from the text, a multiple choice question which states that
Returning to 2017, I notice that Fawn Nguyen has announced that she has a new blog, called "Between 2 Numbers." That's interesting -- on the day I post the Betweenness Theorem, Nguyen posts about a blog that's all about betweenness:
http://www.between2numbers.com/
Nguyen writes that she's still a middle school teacher, except that now she's teaching seventh and eighth grades (as opposed to her usual sixth and eighth grades).
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