Alexa Ortiz Harris Greenwood Felice Mueller Daniel Jones Conical Geometry Conical geometry is an interesting concept that requires some imagination and critical thinking. First of all, we must think about the surface of a cone intrinsically and develop and agree upon specific definitions of some basic notions, before discussing any of the properties of conical geometry. We shall focus on cones with a cone angle less than 360°. Specifically, we will focus on cones with a cone angle less than 90°. In our investigation, we examine the nature of geodesics, Euclid’s Postulates, the properties or behavior of geodesics, circles on the surface, triangles on the surface & the nature of interior angles sums of those triangles. Additionally, we discuss some observations and ideas regarding triangle congruence theorems and the parallel postulates. First, let us define what constitutes a straight line on the surface of a cone. There are two types of geodesics on a cone. The first type of geodesic is called a generator line. The generator is the line (or ray) that begins at the top point, or apex of the cone, and continues infinitely down the side of the cone, perpendicular to the “base”. As we recall, the surface of a cone continues infinitely, such that an actual “base” does not really exist. It was determined that since all generator lines share a single point that is the apex of the cone, they continue infinitely down the other “opposite” side of the cone. The second type of geodesic will be referred to as an up­
and­over line, which can be observed by wrapping a ribbon around any portion of the cone, such that the ribbon lies flush with the surface. The up‐and‐over line goes up the cone, reaches a peak altitude, and then continues its path back down the cone. An example of a the general shape of an up‐and‐over geodesic is given here: However, when an up and over line hits the apex of the cone, it turns into a generator line and begins a path down the cone, perpendicularly to the “base”. Euclid’s Postulates proved to be a bit challenging, so we give a table of our findings, with explanations and discussion following. Incidence A: At least 1 line between 2 points √ Cone 1 Incidence B: At most 1 line between 2 points X Cone 1 Ruler A: travel infinitely forwards and backwards √ Cone 1 Ruler B: never pass the same point twice X Cone 1 Protractor A: At least one line through any pt in any direction √ All Cones Protractor B: At most one line through any pt in any direction X Cone 1 Halfplane A: Cut along the surface of a line, & get exactly 2 pieces X Cone 1 Halfplane B1: There is a line segment from x to y in H √ Cone 1 Halfplane B2: Every line segment from x to y is contained in H X Cone 1 Mirror: Local reflection across every line √ Cone 1 Mirror: Global reflection across every line X Cone 1 We concluded that Incidence A is true because any two points can be connected by some line. However, Incidence B is false, because a generator line and an up‐and‐over line can both pass through two points that are on the cone. Therefore, there is at least, but not at most, one line between two points. This is demonstrated in the picture below: It should be noted that Ruler A holds true along both generator lines and up and over lines. We defined generator lines as lines that continue past the cone point, on to the opposite side, instead of ending abruptly at the apex. Ruler B was determined to be false, because up‐and‐over lines will intersect themselves at least once. For cones that are less that 90°, like the one we used, up‐and‐over lines will intersect themselves at least twice. After thorough consideration, we concluded that both Protractor A is true. There certainly is at least one line through any point in any direction. However, Protractor B is false. Considering the apex of the cone as our point, if an up and over line hits that point, it turns into a generator and goes straight down the cone. Therefore, a true generator and an up and over that turns into a generator both hit the same point and can travel in the same direction and there is not at most one line through any point in any direction. Halfplane A is only true for generator lines and therefore false. Since we defined generator lines as passing through the apex and continuing on the opposite side, cutting along any of these lines will result in two pieces. Halfplane A is false for up‐and‐over lines. Upon consideration of a particular up‐and‐over line on our cone, we determined that cutting along that line results in five different pieces. Therefore, we know that there exist some lines that, when cut along, will result in more than two pieces. Next we will consider Halfplane B1. While an up‐and‐over line will give more than one piece, it is still applicable. If we take some piece H that is the result of being cut along an up and over line or a generator line, there does exist a line in H between x & y. We notice that Halfplane B2 is false on the cone because not every line segment between x & y is contained in H. An up‐and‐over line could essentially “pop out” of H after hitting the point x and then return to the area H before hitting point y. This logic can be extended for cutting along a generator line, and thus, is false. Mirror A is considered to be true. One “weird” point of reflection to consider would be mirroring about a point where an up and over line intersects itself. By reflecting across a point of intersection, it seems logical that a 180° “flip” would occur perpendicularly to the point. As seen here: A more “normal” local reflection would be about some point on L that is not an intersection of itself. This type of mirroring would be the same as any other local mirroring we have encountered. We believe Mirror B is false on the up and over lines. Generators are the only lines that would hold for Mirror B. Since this doesn’t hold for up‐and‐over lines, we can say that Mirror B is false. Upon investigating the properties of straight lines, we considered the questions of whether or not two points are connected by a unique straight line and whether or not straight lines intersect themselves. After some discussion and using a physical model, we noticed that there is not a unique straight line joining two points. In fact, there exists more than one line that joins two points. Generator lines and up‐and‐over lines can both connect two points, such as we found during our consideration of the Incidence axioms. Regarding the question of whether or not straight lines intersect themselves, we immediately observed that up‐and‐over lines always intersect at least twice, as noted in our previous definition, and in our discussion of the Ruler B axiom. When using the physical cone model, we observed that one such up‐and‐over line intersected itself four times. It should be noted, however, that generator lines do not intersect themselves. Triangles on a cone proved to be quite interesting. In general, a triangle is composed of three line segments, and three vertices. Triangles could be formed with two generators & one up‐and‐over line, one generator & two up‐and‐over lines, or three up‐and‐over lines. However, triangles could also be formed with only 2 distinct lines and three segments. Because an up and over line intersects itself, we can use two separate segments of one up and over line paired with a segment of either another up and over or a generator line. Triangles had the potential to be quite “weird” in conical space. The majority of the triangles we found have an angle sum of 180°, but we also observed some cases where the angle sum was greater than 180°. During our investigation, we stumbled upon a few different types of “weird triangles”. We discovered that one type of “weird” triangle could be formed using only one up‐and‐over line. This appears to be a triangle, since there are three line segments, but there are only two vertices. Another “weird” triangle can be found by the “exterior” of a triangle. This is still the same three line segments and three vertices as any “normal” triangle, but is the outside of the triangle instead of the exterior. This will create a triangle with an angle sum greater than 180°. Another type of weird triangle we found was an overlapping triangle, where the triangle overlaps itself. When we map a triangle from the Euclidean plane onto the cone, we observe that, if large enough, it is possible for it to overlap itself. This triangle would have an angle sum of 180°, because it is mapped from the Euclidean plane, although it is possible that there exist some overlapping triangles with angle sums greater than 180°. Another “weird” triangle with an angle sum greater than 180° was observed in two triangles, similar to the “witch hat” triangles found on a sphere. Both the “interior” and “exterior” triangles have angle sums greater than 180°. We discovered that angle sums of a triangle are never greater than 540°, except for exterior triangles. While thinking about the properties of triangles on a cone, we began to wonder if any—or how many—of the triangle congruence theorems would hold on this space. Understanding the nature of the Triangle Congruence Theorems on a cone seemed to be a bit challenging, so we include a table that gives a comparison of Euclidean space and conical space. Theorem Euclidean Space Conical Space Example SSS True False Cone 5 SAS True False Cone 5 ASA True False Cone 5 AAS True ? ______ AAA False False Cone 1 ASS False False Cone 1 In general, we observed that the theorems that failed on the Euclidean plane also failed on the cone. The Euclidean Plane can be mapped to the cone. Taking a triangle on a piece of paper and wrapping it around the cone easily demonstrates this notion. Thus, this implies that if a theorem is false‐‐such as ASS and AAA‐‐on the Euclidean plane, then it is also false when mapped to the cone. To disprove SSS, we can have a triangle with angle sum greater than 180°, and each leg can be rearranged to form a “normal triangle” instead of a “weird” triangle where the interior angle sum would be equal to 180°. This can also be disproven with a “witch hat” type triangle on a cone. A “witch hat” gives two triangles, all with the same sides. This logic also works for an interior or exterior of any triangle. All the sides are the same, but the interior and exterior are not the same triangles. When examining SAS, we noticed that, similar to disproving SAS with a witch hat on a sphere, you could have either a “weird” triangle or a regular triangle. So, SAS is false on conical space. We determined that ASA is also false. This is because a line on conical space wraps around the cone, so it has the potential to intersect our side 1 at two points. Because of this behavior, the angle can remain the same while the other angle and the other side also remain the same. It should be noted that we could not definitively disprove AAS, but our examination of the Triangle Congruency Theorems gave way to a question. We know that for a triangle on Euclidean space, S1+S2 > S3. Is this true on a cone? The answer is, “no, not necessarily. S1 + S2 does not need to be greater than S3!” There can be one long line that travels “up and over” the cone, and two small segments connecting the up‐and‐over to itself, before the up‐and‐over intersects itself. This would mean that this particular triangle would have components of either: one large up‐and‐over line with one small up‐
and‐over line and one small generator line, or one large up‐and‐over line with two small up‐and‐over lines. Although holonomy was not discussed in too much detail, we believe that the holonomy of a triangle on a cone is equal to zero, because it is possible that conical space is conformal to the Euclidean plane. Even if they are not conformal, it is still easily observed that a triangle on the Euclidean plane can be mapped to a cone. Because the holonomy of a triangle on the Euclidean space is 0, it seems logical that the triangles’ holonomy would not be altered by placing it on some different surface. Euclid defines a circle as, “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 are equal to one another; and the point is called the center of the circle.” The circles we observed on the cone varied in appearance. For “small” circles, the appearance closely resembled that of the circles we are used to observing on the Euclidean plane. We also noticed that circles could be formed by “slicing” straight through the cone with a plane, which would theoretically form something similar to a “base” of a cone or a ring around a cone. This is similar to what would be seen if we were to dip the cone in water in a motion such that the altitude line of the cone was perpendicular to the surface of the water. This creates a circle around the cone, where the altitude line would be the center. These circles are equivalent to the latitudinal circles observed on a sphere—that is, they are not great circles, and therefore not straight lines. Another type of circle we found could be considered a “weird” circle. This weird circle is a very large circle that overlaps by intersecting itself when we map it to the surface of a cone. In other words, when we take a large circle on Euclidean plane and wrap it around the cone, two portions of that circle overlap each other. Circles on cones were harder to explore because we could not find a compass that could adapt well to the quirks of conical space. Certainly there exist more interesting circles. One could place a compass inside of a cone and create a circle with some point a little off the altitude line and see what happens. One could also try flopping a circle over the apex of the cone. There is much left to be explored here. When we consider the properties of parallel lines on a cone, we notice some interesting facts about the Parallel Postulates. For example, we observe that parallel lines do in fact intersect on the cone. In other words, the High School Definition of parallel is false for conical space. Two parallel lines do in fact intersect each other, since all lines must intersect at least once as they travel down the cone. Another observation regarding straight lines on the cone is that all generator lines intersect each other at the apex of the cone. From these observations, we were able to define parallel lines on a cone using Playfair’s Postulate and Euclid’s Fifth Postulate. Playfair’s Postulate states, “Given a line L and a point P not on L, there is at most one line L’ through P that does not intersect L.” The words, “at most one line,” allow Playfair’s Postulate to be true on a cone, since we have zero lines that intersect and zero falls under the category of “at most one.” Euclid’s Fifth Postulate states, “If a straight line intersecting two straight lines makes the interior angles on the same side less then two right angles, then the two lines (if extended indefinitely) will meet on that side on which the angles are less than two right angles.” Generally, we noticed that the notion of intersecting parallel lines more closely follows the idea of parallel transport. That is, we noticed that generator lines are parallel transports because they all intersect the “base” of the cone at 90°. To sum up our findings on parallel lines on cones, generator lines and up‐and‐over lines follow Playfair’s Postulate, because they all intersect, and Euclid’s Fifth Postulate, because both intersect the “base” of the cone at perpendicular angles. However, neither generator lines nor up‐and‐over lines follow the High School Postulate. Essentially, the common connotation of “parallel” loses some meaning, as we must choose postulates in order to form a definition of parallel on a cone. The discoveries that resulted from our examination of conical geometry ranged from expected to completely surprising. Nearly half of Euclid’s postulates break down on the cone. The behavior of straight lines change drastically when mapping from the Euclidean plane to the cone, as well as the properties of triangles. Additionally, multiple types of triangles exist on the cone. Almost all of the Triangle Congruency Theorems have been disproved on the cone, with the exception of AAS, which has not been proven to be true, but could not be disproven, as we could not find a counterexample. There was so much to consider during our examination of conical space, we believe that some ideas have the potential for further expansion. For example, the number of different circles on a cone seemed to be the most promising—the variations seemed vast. There are still so many questions that remain to be asked and answered. How many types of quadrilaterals are there? How many types of any n‐gon are there? How do you calculate the area of a weird triangle? What is calculus like on a cone? If you take a circle on a cone and spin it around a diameter line, will it still create a sphere? How does this sphere realize itself in conical space? What would geometry be like on that sphere? There are still so many exciting questions to be asked. It’s hard to not get carried away thinking about the incredible beauty and endless possibilities of conical geometry.