Internal Assessment Simon KimCarson Graham SecondaryMs. J. Dai2017Proof of Gabriel’s Horn (Torricelli’s trumpet)Does the Mathematical paradox of Gabriel’s Horn affect its existence?ContentsIntroduction The figure of Gabriel’s HornGeometryCalculations VolumeSurface AreaSurface Area Formula DerivationApplication of Surface Area FormulaImproper Integrals ExplanationComparing Improper IntegralsSurface Area of Gabriel’s HornResults Analysis Conclusion{equation number} – Reference number for equationsFigure x.y – Reference number for graphs, data, and other figures1) Introduction Gabriel’s Horn or Torricelli’s Trumpet is a unique geometric figure due to its interesting properties. It is known for being discovered and studied by Evangelista Torricelli(15 October 1608 – 25 October 1647), an Italian physicist and mathematician renowned for his creation of the mercury barometer and Torricelli’s equation(Britannica, 2017). This figure has been titled alluding to the Archangel Gabriel’s Horn he blows on Judgement Day, symbolizing the connection between finite and infinite(Theindeliblelifeofme, 2014). The property that causes it to differ from others, is that it has infinite surface area, but has finite volume. Many have disputed(C. Wijeratne, 2015), claiming that there are paradoxes with the concept of Gabriel’s Horn and cannot possibly exist.I have always been intrigued by different topics of debate, and Gabriel’s Horn has especially peaked my interest due to its paradoxical nature. While growing up, I often watched television with my parents, and often times we would watch American news where they would frequently broadcast debates. Although I probably lacked the knowledge at the time of what topic the debates were on, I always cheered on one side and wished ill on the opposing party. Regarding Gabriel’s Horn, I did not believe a shape of finite volume and infinite surface area could possibly exist. I was very cemented in my belief during the early stages of my research, as I came across convincing paradoxes that supposedly disproved the existence of Gabriel’s Horn. The paradoxes and my mistake was relating a theoretical figure to a physical idea of possible figures. However this did not sway me and this topic is worth exploring because investigating this figure of infinite length will help with the understanding of infinity, disprove myths and paradoxes, and possibly look for applications.Therefore my goal for this research is to discover how a mathematically paradoxical figure can exist, as well as possible applications for Gabriel’s Horn, to find possible uses for the convenience of other maths, similar to the function of complex numbers such as i. I will obtain the data for this research by looking and using multiple past research stemming from Torricelli’s investigations majorly second hand sources because Torricelli’s work was written in Italian. Therefore, I believe reading translated secondary sources will be more efficient rather than attempting to translate the primary source myself. Even if some formulae may be evident in the primary source, I would not be able to fully understand the context. Furthermore, I will derive the volume and surface area myself, to understand how this mathematically and seemingly logically defying figure can exist.2) The figure of Gabriel’s HornGeometry Gabriel’s Horn is a solid of revolution that is created using the following properties. Figure 2.1 First, the function of y=1xis given. Figure 2.2 Second, the domain is limited to x1 (area is represented in red) Figure 2.3 Third, function is revolved around the x axis when 1 x +,resulting in a cone shape, which then forms the shape of a funnel Therefore this is the final shape of Gabriel’s Horn, with dimensions of fx=1x revolved around the x axis with the domain of 1 x +.Figure 2.4 The volume of the disks are r2hwhich is the formula for calculation of a cylinder. (represented in yellow) from Stewart, J. (2003).In order to find the volume of any solid of revolution, the volume of all disks(refer to Figure 2.4) must be found and then summed from start point a to endpoint b. In the case of Gabriel’s Horn a=1and b=. Logically, if a shape has an endpoint of infinity, one can only assume the volume is also infinite, but after discovering that this was not the case and that the volume was indeed limited, this peaked my interest and I am deciding to prove the finite volume of Gabriel’s Horn utilizing knowledge within the math HL syllabus.Similarly, surface area should also be infinite from an endless endpoint, but with the knowledge that volume is indeed not infinite, the surface area could be anything. Therefore, I looked to prove or find evidence if the surface area was infinite or otherwise.3) Calculations Variables, formulae and terminology clarifications Volume of cylinder V r2h Where r= radius h= height a integrated to b ab{1} Surface Area of a disk(refer to Figure 2.4) C 2r Where r= radius Centroid meana) VolumeTo start, we have our area formula for 1x from point x=1integrated to x=. 11xdxNext, we obtain V = volume of a solid of revolution by using the general formula found within the math HL syllabus. V=h0x=ax=bfx2h (Since the volume of a cylinder is r2h, we use the function of x for r, where r= radius of cylinder)We then convert the 2 dimensional area formula for 1xinto the general formula for the volume of a solid of revolution V=h0x=1x=1x2h Which simplifies to V=11x2dx Then, V=a1a1x2dx We convert 1 to a1a for calculation purposes, as is not a number For fx=1×2 , We use the power rule… xndx=xn+1n+1 Where n = -2…to find the integral Fx= x-1-1 = -1x V=a-1×1 a We use abecause we can’t use directly as is not a numberAnd we continue by substitute x for 1 and a, V=a-1a–11 =a-1a+1 =-0+1 Since -1a converges to 0 as a =1 V=?The volume of Gabriel’s Horn is . From the calculations above, I have concluded that the volume of Gabriel’s Horn isunits3, which contradicted my original hypothesis that a solid with infinite distance between a and b. I was surprised at this fact initially, but after some thought on the process of the creation of Gabriel’s Horn, I realized that it would be flawed to compare a theoretical figure to physically possible shapes. b) Surface AreaIn order to find the surface area of Gabriel’s Horn we must first derive a formula for the surface area of a solid of revolution because this knowledge is not found within the math HL syllabus. We first use the formula for the circumference of the disks {reference equation 1} C=2r Where r = radiusWe then substitute our function 1xfor r C=21xi) Surface Area Formula DerivationIn the case of Gabriel’s Horn, I hypothesized that surface area would be finite, due to my prior derivation of the finite volume.Since the disks {reference equation 1} have a circumference of21x, we would then need to multiply that value by the length of the solid.This would give us the total surface area of Gabriel’s Horn. This is generalized in Pappus’ 1st Centroid Theorem, in which it describes all solids of revolutions and not limited to Gabriel’s Horn. Sa=?d Where Sa= Surface Area ?= Arc Length d=Distance travelled by the solid of revolutions geometric centroidA centroid is defined as the geometric center or arithmetic mean position of all points in a shape(Bourke, 1988). In this instance, we are finding surface area of a solid of revolution that revolves around the x-axis. Therefore, the centroid travels at y = 0 for all values of 1 x +, and fully revolves around the x-axis for 21x.This can be shown as d=21x1dxFor the length between 2 points Pi-1,Pi for which we can find the arc length ?=ci=1cPi-1,PiSay yi=yi-yi-1=fxi-fxi-1The Mean Value Theorem, which is derived from Rolle’s Theorem, which is in turn derived from the Extreme Value Theorem and Fermat’s Theorem, states that there is a point xi*on the interval xi-1,xi fxi-fxi-1=f’xi*xi,xi-1 yi=f’xi*x fxi-fxi-1 is equal to yi and xi,xi-1=xThe length is Pi-1,Pi=xi,xi-12+yi,yi-12 =x2+f’xi*2×2 =1+f’xi*2×2 =1+f’xi*2 x? The arc length is ?=ci=1cPi-1,Pi ?=ci=1c1+f ‘x2 x Which can be rewritten using knowledge from definite integrals ?=ab1+f ‘x2 dxTherefore we can use the surface area formula Sa=?d Sa=2rab1+f ‘x2 dx Where1+f ‘x2= ?= Arc length 21x1dx = d= Distance travelledii) Application of Surface Area FormulaTo find the surface area of Gabriel’s Horn, we substitute our r variable with our function 1xonce moreSa=21xab1+f ‘x2 dx =2b1b1x1+f ‘x2 dxWe then use our f ‘x of fx=1x, When fx=1x, f ‘x=-dydxxx2 = -1x2To substitute f ‘x below Sa=2b1b1x1+-1×22 dx =2b1b1x1+1×4 dx =211×1+1×4 dxiii) Improper Integrals ExplanationFor the term of 1+1×4, all values must be >1 1= 1 ? 1+x> 1 Adding any positive value x will cause it to be greater than 1 because x4is always positive ? 1+1×4>1 ? 1×1+1×4>1×1 Adding 1xto both sides of the inequality ? 11×1+1x4dx>11x1dx Adding the improper integral 1 to both sides of the inequality{2} ?211×1+1x4dx>211x1dx Adding 2to both sides of the inequalityiv) Comparing Improper IntegralsQuestion #9 from section 22E.2 of the HL coursework(D.Martin, 2012) states that “the … area from x=1 to infinity is infinite for the function y=1x.” This is shown as A=11xdx = If this were to be proven true, then 211x1dx= would be true as well because 11xdx < 211x1dxTherefore, we need to prove that A=11xdx is infinite{3} A=11xdx =b1b1xdx We use bbecause we can't use directly as is not a number =blnx|1b Because for fx=1x , f'x=lnx =b(lnb-ln1) =blnb ln1 is irrelevant if b{4} A=v) Surface Area of Gabriel's HornSince 11xdx < 211x1dx < 211x1+1x4dx , {reference equation 2, 3} < 211x1+1x4dx?211x1+1x4dxmust also be ? Surface Area of Gabriel's Horn is 4) Results Analysis The results I got from calculating Gabriel's Horn were fascinating, due to the fact that I had no idea that this solid was possible, and it seemed to contradict the basic common sense that if something is infinitely long, it must hold infinite volume. Furthermore, as previously stated, in the course work of the math HL work within the textbook(D.Martin, 2012), I noticed that it stated that "We call thisthe relationship between the function of 1x where x1and Gabriel's horn a mathematical paradox." This statement is correct, proven in {reference equation 4}, proving the relationship of 1x where x1and Gabriel's horn is indeed mathematically paradoxical, however the fact that states the Gabriel's Horn does is exist is not paradoxical, concluded from calculations and evidence gathered in section 3) Calculations. A well-known issue with Gabriel's Horn is that of the Painter's Paradox, in which it states that since there is a volume , while having the surface area of Sa=, if it were to be filled with units of paint, Gabriel's Horn would be completely filled, however, when you remove the paint, the surface area (the inner surface area is identical to the outer surface area), should be completely coated, due to it being filled, but also should not, due to the surface area being infinite. The mistake/misinterpretation made here is attempting to contextualize Gabriel's Horn in the physical world. It cannot exist in the real world, as it would have to stretch to infinity. Furthermore, if we were to assume it could exist in physical form, there are new issues that arise. For example, there would be a point where paint molecules would no longer be able to continue down Gabriel's Horn, as the passage would become so narrow to the extent where atoms would not fit. If we were to assume paint could be filled, it would take an infinite amount of time for the paint to fill. My originality in this exploration comes from my lack of understanding of why the formulas; Volume of a solid of revolution, Surface area of a solid of revolution. During my research, I found that many articles and journals have previously found the results of Gabriel's Horn as I did, coming to the conclusion of finite volume and infinite surface area. However, none of these articles or journals explain the different elements of prior knowledge needed to understand their given equations. For example, in Chanakya's On Painter's Paradox: Contextual and Mathematical Approaches to Infinity article, the proof of the surface area of Gabriel's Horn is shown as a1a21x1+-1x22dx a1a21xdx = a1a2ln x1a=a2ln a-ln 1=The article does not elaborate on where these equations come from, neither does it explain what the original formula derives from. Similar to this case, the other articles and journals on Gabriel's Horn assume the reader has an extensive prior knowledge of calculus and geometry. As a reader without much knowledge(prior to this exploration) of integral calculus, I was confused as to how all these symbols and numbers came together to create a representation of infinite surface area. It was through my own derivation through understanding found within the Math HL syllabus that led me to realize where the formula for volume and surface area of a solid of revolution comes from. During my exploration of finding surface area for solids of revolution, I found the following general formula Sa=2rab1+f 'x2 dx This formula is derived from a combination of the arc length formula and the Centroid Theorem.• The Centroid theorem is derived from knowledge of centroids and the Pythagorean Theorem• The arc length formula is derived from knowledge of functions and the Mean Value Theorem• The Mean Value Theorem is derived from knowledge of differential calculus and Rolle's Theorem • The Rolle's Theorem is derived the knowledge of functions, the Extreme Value Theorem and Fermat's Theorem • The Extreme Value Theorem is derived from knowledge of functions • Fermat's Theorem is derived from knowledge of differential calculus, mathematical induction and inequalitiesAll of the theorems needed to prove arc length ?=ab1+f 'x2 dx all required mathematical knowledge much higher than the content within the math HL syllabus. I could not find a communicatively efficient way to show proof for the arc length formula. Therefore, I summarized the theorems as simply as I could. My final results were as followsGabriel's Horn, a solid of revolution is created by revolving the function of 1xat x=0 for the domain of 1xGabriel's Horn has a finite volume of Gabriel's Horn has an infinite surface area Arc length can be calculated with the formula ?=1+f 'x2 Arc length can be derived using knowledge from the Mean Value TheoremMean Value Theorem can be derived using knowledge from Rolle's TheoremRolle's Theorem can be derived using knowledge from Fermat's Theorem, the Extreme Value Theorem and application of differential calculus in finding the properties of curvesSurface area can be found through Pappus' 1st Centroid Theorem Sa=?d, which for the case of a solid of revolution is Sa=21xab1+f 'x2 dxContradictions and paradoxes appear only when attempting to conceptualize and apply Gabriel's Horn in physical form.5) Conclusion The purpose of my investigation was to explore if an apparent mathematically paradoxical, physically impossible figure could exist, and to calculate it's dimensions. Since this figure is purely mathematical, and is unable to be created in the real world, it's applications are limited to helping further our understanding of infinity, as well as other mathematical concepts. Overall, I managed to disprove my original thinking of Gabriel's Horn not being able to exist by deriving the surface area and volume of this solid of revolution myself. Furthermore, I was able to find reasoning behind the paradoxes that supposedly opposed the existence of Gabriel's Horn. A similar shape that caught my interest was a shape known as Koch's Snowflake. This is a fractal with the dimensions of area=235x2 where x = side length of the original triangleperimeter= This is another figure that has finite and infinite attributes. The interesting matter here is that in Koch's Snowflake, the 1 dimensional attribute(perimeter) is infinite, whereas the 2 dimensional attribute(area) is finite. In the case of Gabriel's Horn, the 2 dimensional attribute(surface area) is infinite, and the 3 dimensional attribute(volume) is finite. This led me to question whether this pattern could continue to past our 3rd dimensional world. What if there was a dimensional figure has 4 dimensions or even beyond, and has traits of infinity in the 3rd dimension. This question may be answered with the broadening of the understanding of calculus and other maths. The math I have used to derive the different formulas are not limited in any way, as they have been proven multiple times, in research papers, journals and math papers. In conclusion, the fact that the function to create Gabriel's Horn has infinite area, but once revolved around the x-axis gives a volume of is indeed a mathematical paradox, but this does not prove that the Gabriel's Horn cannot exist. I have used my research and exploration to prove that Gabriel's Horn does exist, which leads to the result of; Gabriel's Horn does exist, and the relationship between Gabriel's Horn and the function necessary to create it 1xis indeed paradoxical. Works CitedBourke, P. (1988). Calculating the area and centroid of a polygon. Swinburne Univ. of Technology.Stewart, J. (2003). Single variable calculus: Early transcendentals. Belmont, CA: Thomson Brooks/Cole.Goodman, A. W., and Gary Goodman. "Generalizations of the Theorems of Pappus." The American Mathematical Monthly, vol. 76, no. 4, 1969, pp. 355–366. JSTOR, JSTOR, www.jstor.org/stable/2316426.Wijeratne, Chanakya & Zazkis, Rina. (2015). On Painter's Paradox: Contextual and Mathematical Approaches to Infinity. International Journal of Research in Undergraduate Mathematics Education. . 10.1007/s40753-015-0004-z. Wijeratne, C., (2015) Infinity, Paradoxes and Context. Phd Dissertation. Simon Fraser University. summit.sfu.ca/item/14933Martin, D. (2012). Mathematics for the international student: Mathematics HL (Core). Haese Mathematics.Dawkins, P. (2003). Calculus 1. Retrieved December 17, 2017, from http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspxThe Editors of Encyclopædia Britannica. (2017, January 13). Evangelista Torricelli. Retrieved December 14, 2017, from https://www.britannica.com/biography/Evangelista-TorricelliMancosu, P., & Vailati, E. (1991). Torricelli's infinitely long solid and its philosophical reception in the seventeenth century. Isis, 82(1), 50-70.Examples of explorations. (n.d.). Retrieved December 17, 2017, from https://ibpublishing.ibo.org/live-exist/rest/app/tsm.xql?doc=d_5_matsl_tsm_1205_1_e&part=2&chapter=2