Pitching a Curve Dr. Maurice Burke Montana State University- Bozeman October 19, 2017 What is a circle? Let’s zoom back 5000 years to 3000 BC.…… The time of the first pharaohs and humanity’s first writing in river cultures…… the birthday of the old tree on the right. Mammoths still walked the earth and Ötzi, the iceman, was hunting in the alps! What did people then think a circle was???? Quiz: If C=3, is A >1 ? Here are two “Biscuits” baked about 2000 BC by some Babylonian students. What do the numbers mean? The circumference is 3 and the area is 45/60 or ¾. The Bab classic “What’s the area of a tree stump’s circular top” problem “1 40 mu-uh-hi i-s:´ı-im ˇsu-ul-li-iˇs-ma = 5 ki-pa- at i-s:´ı-im i-l´ı = 5 ˇsu-ta-ki-il-ma 25 i-l´i = 25 a- na 5 i-gi-gub ´e-em i-ˇsi-ma = 2 05 A.ˇS`A i-l´I” Translation: “Triple 1;40, (the diameter of) the top of the log, and 5, the circumference of the log, will come up. Square 5 and 25 will come up. Multiply 25 by 0;05, the coefficient, and 2;05, the area, will come up. So it appears : • C = 3xD….. To us: C=πD • So their “π” was 3. But they had no theory of π - Just a pattern they noticed in circular regions. 퐶2 • To us, 퐴 = . To Babylonians, 4휋 2 퐶 1 2 퐴 = = × 퐶 푠표 4×3 12 5 2 2 A= × 퐶 = 0; 05 × 퐶 ) 60 • Hence, the coefficient 0;05 THERE IS NOTHING WRONG WITH THIS!!! I wonder if we asked students to come up with a formula for the area of an ellipse given its circumference and diameter….. Here is a related Thought: Oldest Mathematics Exercises on Record, over 5000 years old! • Tablet shows two exercises on calculating the area of quadrilateral fields. Contrived numbers reveal them as school exercises. • What fascinates me is that the Surveyor Formula was used for Sumerian ca. 3200 BCE. Uruk over 2000 years in Egypt and Tablet W 19408 Mesopotamia!!!! Surveyor Formula Eleanor Robson says: “The Old Babylonian circle was a figure—like all OB geometrical figures—conceptualized from the outside in. In such a situation, there could be no notion of measurable angle in the Old Babylonian period…. The radius was never conceptualized as a rotatable line.…The area of a circle is never calculated directly from its diameter… Both the circle and the circumference are called in Old Babylonian kippatum from the verb kapapum ‘to curve’. The diameter was necessary in order to conceive of a circumference or circle as a loop whose opposite points are all equidistant.” To use Linnaean terminology, their conceptual schema of circle, which includes its properties, major components and their relations, is a different species than our schema, but clearly part of the same genus. It is structurally different in fundamental way from our schema. Nonetheless, it has coherence and use. Fast Forward 2000 years 600 BC – 200 BC, The Greek Miracle 1. Definitions: Outgrowth of philosophical Arguing? 2. Theoretical discrimination of Number and Magnitude 3. Discovery, Exploration, and Development of “Circle”, and only a few (6) other curves, using synthetic geometry. No theory of “curves”. They are still tools. 4. Curves are physical, kinematic, stereometric and emphatically not numerical. They are “Continuous.” and that means unbroken, their parts touch; Locus is not a “set of points” but the place where points reside; points do not touch each other, they are not “parts” of a curve. So curve is more than a set of points. AHHHH ZENO! Euclid’s Definitions (~300 BC) • Def. 2.A line is breadthless length. (Not a set of points! It’s a curve!) • Def. 4.A straight line is a line which lies evenly with the points on itself. (It is not the same as its points, they lie on it. Two points determine a line???) [Other defs: A plane figure is a surface with a boundary] • Def. 15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. (A circle is a region and the curve containing it!) [Evidence of evolution of schema: Refinement of structure] Greeks: Curves are initially used to solve problems that resist circles and lines. • Trisection Problem (Quadratrix of Hippias - The oldest curve in mathematics next to the straight line and circle ~450 BC) • Duplicating the Cube (Conics) Quadratrix • Kinematically defined • Determined by the motions of two points (yellow dots in figure) • Used for Trisecting Angle • Constructed point by point with ruler and compass Your students can do these simulations Use GeoGebra….Now on Cell Phones! Let’s try it. The Delian Problem and Conics • Doubling the Cube: Construct the side of a cube whose volume is twice the volume of a given cube. So, if given cube has volume 1, then you must construct segment of length so that the cube with side length x would have a volume of 2. I. e. , 푥3 = 2, or 푥 = 3 2. (Perhaps the most consequential problem from antiquity.) • Hippocrates (~450 BC) discovers that you can do this if you can construct two mean proportionals between segment a and a segment twice as long as a. I.e. construct segments x and y such that 푎 푥 푦 a:x = x:y = y:2a or = = 푥 푦 2푎 • 푊푒 푤푙푙 푠푒푒 푛 푎 푚표푚푒푛푡 ℎ표푤 푡ℎ푠 푤표푟푘푠. The conic sections are discovered Archytas (~400 BC), friend of Plato, teacher of Eudoxus (~380 BC), who is teacher of Menaechmus (~ 350 BC) who is tutor of Alexander the Great (~340 BC), shows how to find two mean proportionals between segments a and b by intersecting a cone with a cylinder with a torus, all surfaces defined by revolving lines and circles. Menaechmus simplifies the construction to the intersection of two planar sections of cones (Conics discovered!) Section of Right-Angled Cone Section of Acute-Angled Cone Exploration and Development of Conics • Euclid book Conics (now lost) • Archimedes (~250 BC) intersects “sections of right-angled cones” with “sections of obtuse-angled cones” to find where to section a sphere so that the volume of the two pieces have a some given ratio to each other. • E.g., where would you slice a sphere of radius 1 so that you cut off a third of its volume? Arch’s method reduces this to 1 finding h so that 4: ℎ2 = (3 − ℎ): . 4 • This, to us, is an equation ℎ3 + 1 = 3ℎ2. BUT, Arch was solving a proportion between magnitudes by a construction, ours is solving an equation of numbers by using algebraic/numeric operations. Apollonius: Theory of Conic Curves: Their formal baptism • Introduces language: absicssa, ordinate, parabola, hyperbola, ellipse. Considers the curves as a single general family rather than as three unrelated curves. • Defines conics stereometrically and uses the most general 3-D model, the oblique cone, and double-naped cone for his initial definitions and theorems. • His language is still geometric where square means a square region and not some number. He uses similar triangles and the Fundamental Property of Circles When two chords intersect inside a circle, the products of the segments of one chord equals the product of the segments of the other chord. Corollary: If one chord is a diameter and is perpendicular to the other, then it bisects the other chord. Apollonius Parabola Apollonius Parabola GeoGebra let’s us explore the conics and visualize the arguments of Apollonius. In our algebraic way expressing things: 푥2= (4푝)푦 4p is the latus rectum of the parabola, so P is half the size of the gold segment. Your students can do these 3-D diagrams and study the ratios of components using GeoGebra…. So How To Solve 푥3 = 2 (Omar Khayyam’s Construction ~1100 AD) 1:x = x:y = y:2 1 푥 푦 i.e. = = 푥 푦 2 Find x and y We have 푦 = 푥2 So, 푦2 = 푥4 But we have 푦2 = 2푥. SO: 푥4 = 2푥 표푟 푥3 = 2 The Greek Legacy on Conics (i.e. The mindset for the next 2000 years!!) • There are no negative coordinates, let alone numerical coordinates. No numberline! • No equations of number and numerical operations, but proportions between magnitudes, given or constructed. Not easy to work with – you can’t even cross multiply! • The auxiliary lines are imposed on the curve after the curve is given and do not form a coordinate frame of reference independent of the curve. • The curve always passes through a point called origin. • Greek conceptual schema for curves is a different species, but of same genus as ours. Again, it is structurally different than ours. Fast Forward 1500 Years (~1100 AD) Evolution never stops! • Arabian Scholars inherit the Greek Tradition. But gradually, focus on algebra becomes dominant part of their mathematics. • Algebra is born as a subject of numbers and operations and equations. Equations are used and manipulated as mathematical objects. It is not symbolic, but rhetorical algebra. We find systems of equations and willingness to cross categorical lines in operating on quantity. • The number concept evolves as Arab algebraists start to accept irrational numbers like square roots as solutions and as coefficients of equations. Negative numbers still frowned upon as false (but studied and operated on!!) and no consensus on the status of irrationals as numbers. Just what are these inexpressible things? Never coherently answered! • Very few new curves get studied and coordinate geometry does not exist.