International Journal of Modern Research in Engineering & Management (IJMREM) ||Volume|| 1||Issue|| 3 ||Pages|| 43-51 ||March 2018|| ISSN: 2581-4540
Poincare Conjecture Proven and Disproved Using Imaginary Numbers, Imaginary Manifolds, Always Changing Manifolds, Suns or Stars, Rhombus, Trapezoids, Ellipses, Triangles, Parallelograms, Squares and Word Meanings Mathematical Classifications 03Fxx 01Axx 51Nxx 1,
James T. Struck BA, BS, AA, and MLIS
Affiliation- President A French American Museum of Chicago, President Dinosaurs Trees Religion and Galaxies, Researcher NASA, Assistant Midwest Benefits Group Address- PO BOX 61 Evanston IL 60204
-------------------------------------------------------ABSTRACT--------------------------------------------------A manifold means “1. Having many forms, parts, etc.” [1] Manifolds which have many parts, forms are not necessarily one to one to 3 spheres. Poincare conjecture can be proven and disproven based on the case study of imaginary manifolds, imaginary numbers, ellipses, rhombus, trapezoids, parallelograms, triangles, squares, always changing manifolds and the Sun or stars. Proof and disproof are reasonable for the Poincare Conjecture. Homeomorphic has 2 meanings. One meaning of homeomorphic is similar in form and one meaning of homeomorphic is deformable. [2] Proof and disproof of Poincare Conjecture are supported by the 2 definitions of homeomorphic and the definitions of manifold. Dr. Grigori Perelman’s proof of the Poincare Conjecture does not place enough emphasis on the meaning of homeomorphic as having similarity of form or the definition of manifold as many, diverse or varied. ----------------------------------------------------------------------------------------------------------------------------- ---------Date of Submission: Date, 18 March 2018 Date of Accepted: 27 March 2018 ----------------------------------------------------------------------------------------------------------------------------- ----------
I.
METHODOLOGY
I rely primarily on meanings of homeomorphic and manifolds or linguistic discussion. I also use case study or examples of imaginary numbers, always changing manifolds, imaginary manifolds, rhombus, trapezoids, ellipse and Sun and stars to discuss the Poincare Conjecture. Another type of disproof is that we are not completely sure what Henry Poincare actually saw as the Poincare conjecture as we cannot speak to him directly in 1900 and 1904. The Meaning of Homeomorphic: Discussion of the Poincare Conjecture can be seen as related to the Meaning of homeomorphic. I use a math resource Wolfram Math World to define homeomorphic. Here is Wolfram Math world’s definition accessed on 12/31/2017. “There are two possible definitions: 1. Possessing similarity of form, 2. Continuous, one-to-one, in surjection, and having a continuous inverse.” The most common meaning is possessing intrinsic topological equivalence. Two objects are homeomorphic if they can be deformed into each other by a continuous, invertible mapping. Such a homeomorphism ignores the space in which surfaces are embedded, so the deformation can be completed in a higher dimensional space than the surface was originally embedded. Mirror images are homeomorphic, as are Möbius strip with an even number of half-twists, and Möbius strip with an odd number of half-twists. In category theory terms, homeomorphisms are isomorphisms in the category of topological spaces and continuous maps. [3] Here I rely on both meanings. I use similarity of form and deformable as the definitions of the word homeomorphic in order to show that there can be both proof and disproof of the Poincare conjecture related to those definitions. The Meaning of Manifold: For the meaning of manifold, I rely on the Merriam Webster definition as the common word use or everyday use of manifold is arguably more linked to the meaning used by Henri Poincare at the turn of the 20th century in 1900 and 1904. Here is a meaning of manifold according to the online Merriam Webster dictionary accessed on 12/31/2017. “Definition of manifold
www.ijmrem.com
IJMREM
Page 43
Poincare Conjecture Proven and Disproved Using… 1 a: marked by diversity or variety • performs the manifold duties required of him b: many [4] As the word manifold means many or diverse or variety, based on word definition alone, something that is diverse varied and many is not similar to a 3 sphere. Linguistic Disproof based on diverse, many, varied does not mean 3 sphere is simple to argue for or support. Case studies of Imaginary Manifolds, Imaginary Numbers and Always Changing Manifolds and Sun Imaginary Manifold or Imaginary Number Disproof For a definition of imaginary number, I again rely on Wolfram MathWorld accessed 12/31/2017. Imaginary Number means “Although Descartes originally used the term "imaginary number" to refer to what is today known as a complex number, in standard usage today, "imaginary number" means a complex number that has zero real part (i.e., such that ). For clarity, such numbers are perhaps best referred to as purely imaginary numbers. A (purely) imaginary number can be written as a real number multiplied by the "imaginary unit" i (equal to the square root ), i.e., in the form . [5] An imaginary manifold form can be imagined which does not actually exist other than in imagination. Manifolds that are imaginary would not easily be similar or deformable into an actual 3 sphere. An imaginary number like “i” is not a 3 sphere. Imaginary numbers are manifolds, but imaginary numbers are not similar to 3 spheres. Poincare disproven based on imaginary manifolds and imaginary numbers. As we do recognize imaginary numbers like “i” as existing, we can see that an i or imaginary number is not a 3 sphere and is not similar in form to a 3 sphere. To maintain the meaning of similarity in form, we should be able to say an “i” or imaginary number is not similar to a 3 sphere. Something that is imaginary in form is not similar in form to an objective 3 sphere. A real number multiplied by an imaginary unit is not a 3 sphere. Poincare Conjecture Proved Based on Imaginary Manifolds and Numbers: One still could prove the Poincare conjecture too by showing an imaginary manifold that deforms into a 3 sphere. Such a deformation from imaginary manifold to 3 spheres is based on the ability to imagine or draw or see deformation without cutting from an imaginary number like I into a 3 sphere. The “i” or imaginary number grows on all sides and becomes a 3 sphere. Deformation from I to 3 sphere is a possibility. We take an imaginary manifold or number like i, make it a real manifold and then we deform the manifold into a 3 sphere. Simple growth or expansion can be used for deformation. Proof and disproof were just shown and illustrated with imaginary manifolds. Just expand I into a 3 sphere and the deformation is viewable. Just look at the expansion or growth of I into a 3 sphere and we can see that I and 3 spheres are similar. In the real world, we have imaginary particles or virtual particles. In the real world we have imaginary numbers. I was taught imaginary numbers in Algebra in sophomore year of high school. Imaginary numbers are well known as the square roots of negative numbers. Imaginary numbers are a branch of numbers similar to e’s or pi; important numbers in number theory. An imaginary number can be deformed into a 3 sphere, but is not similar to a 3 sphere. Poincare conjecture is both provable and disproved using imaginary numbers. An imaginary manifold can be seen as not similar to a 3 sphere and not deformable into a 3 sphere. Something imaginary cannot become real arguably. On the other hand, something imaginary like an imaginary manifold could be deformed into a real manifold and then seen as deformable into a 3 sphere. Proof and disproof are both supported by examination of imaginary numbers and imaginary manifolds. Always Changing Manifolds. I introduced this argument for always changing manifolds not being homeomorphic to 3 spheres in July 2010. Always Changing Manifold Disproof: An always changing form can be imagined. Manifolds that are always changing would not be similar to or deformed into a stable 3 sphere. A manifold can always be oscillating, disappearing, transforming, and changing forms. Such a manifold would not be similar or deformable into a stable 3 sphere. Poincare Conjecture can be disproven based on the always changing manifold with varied, diverse, or many forms not being similar or deformable into a stable 3 sphere. Always Changing Manifold Proof: On the other hand, one can have an always changing manifold and an always changing 3 spheres.
www.ijmrem.com
IJMREM
Page 44
Poincare Conjecture Proven and Disproved Using… As a type of proof of Poincare conjecture, always changing manifolds can be similar and deformable into always changing 3 spheres. Even from an always changing manifold, a deformation to stop the changing characteristic or form of manifold would be similar and deformable into a 3 sphere. Sun or Stars Disproof: The Sun also can be described as both similar to a 3 sphere and deformable into a 3 sphere and not similar and not deformable into a 3 sphere. The Sun arguably is a simple compact closed manifold. The Sun has roughly 333,000 times the size of Earth. [6] The Sun with much larger size is not similar to a very small 3 sphere. The Sun is constantly shooting spicules up in the air “at a rate of 20 km/s (or 72,000 km/h) and can reach several thousand kilometers in height before collapsing and fading away.” [7] The Sun which is always changing is not similar to a stable 3 sphere. We could deform the Sun into one stable 3 sphere, but a Sun which is 333,000 times the size of the Earth or always shooting off spicules is not similar in form to a very small 3 sphere. Let’s take all the stars considered as a manifold in comparison to one 3 spheres. In our galaxy alone, there are from 100 to 400 billion stars. [8] If there are 100 to 400 billion stars, those many stars are not similar in form to one 3 spheres. We could argue that each individual star is similar to a sphere, but the collection of all 100 to 400 billion stars in our galaxy would not be directly similar in form to one 3 sphere. The Sun and Stars and their differences in form, appearance, structure, activity from a 3 sphere can be seen as a disproof of the Poincare Conjecture. Exploding stars, binary stars, interacting stars, Supernovae, dying stars are not the same as a 3 sphere. Sun and Stars Proof of Poincare Conjecture: The sun and stars as more numerous and much larger and changing could still be seen as vaguely similar to a 3 sphere. Similar is an arbitrary term. Is something like something else or different? Is a star similar to a 3 sphere or is a star dissimilar than a 3 sphere? One massive star or many numerous stars with changing spicules can be seen still as similar to 3 spheres. Stars and Suns are sometimes like 3 spheres, but Suns and Stars movement makes them different. How about deformability? We do not need to rely on Dr. Grigori Perelman’s discussion of Italian Mathematician Gregorio Ricci and his heat flow work. From Wikipedia.org accessed on December 31, 2017 “Gregorio RicciCurbastro (Italian: [ɡreˈɡɔːrjo ˈrittʃi kurˈbastro]; 12 January 1853 – 6 August 1925) was an Italian mathematician born in Lugo di Romagna. He is most famous as the inventor of tensor calculus, but also published important works in other fields.” [9] Ricci was not working on Poincare Conjecture when he invented the Ricci Flow. As Ricci flow was not intended to be a proof of the Poincare Conjecture as Perelman used the Ricci flow, we cannot be sure the application is entirely appropriate. What did Ricci really intend the Ricci Flow to be used for? That purpose can never be determined due to a lack of evidence how Ricci intended his flow to be used. Aside from the Ricci Flow issue however, can a star be deformed into a 3 sphere? Arguably, yes stars can be deformed into 3 spheres. Also No stars cannot be deformed into 3 spheres as there would need to be folding or cutting of spicules, sunspots and coronal mass ejections. We can imagine many numerous stars and one massive star and Sun as deformable into a 3 sphere. Deformation means changing into without folding or cutting. We could change a Sun into a 3 sphere. We can deform many 200 billion stars in our galaxy into a 3 sphere. Imagination allows these transformations without cutting or folding. Shrinking, deflation, Growth, Inflation ideas would be involved in making a Sun or Star into a small 3 sphere or a very large 3 sphere. We do not need to rely on any kind of flow or Ricci Flow to do deformation!
II.
RHOMBUS AND TRAPEZOID DISCUSSION
A trapezoid is “In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid[1][ Here is an image accessed from Wikipedia.org article on the subject. This trapezoid shape
[10] Is not the same as a 3 sphere! This shape however could be deformed into a 3 sphere. The Poincare Conjecture again easily disproved and provable.
www.ijmrem.com
IJMREM
Page 45
Poincare Conjecture Proven and Disproved Using… Similarly, with a Rhombus. A rhombus is defined as “In Euclidean geometry, a rhombus(◊) (plural rhombi or rhombuses) is a simple (nonself-intersecting) quadrilateral whose four sides all have the same length”
[11] This shape is not a 3 sphere, so the Poincare conjecture is disproved. Rhombus are not 3 sphere. A rhombus is different than a 3 sphere not the same. Could we deform a rhombus into a 3 sphere? We could stretch the sides into a more spherical structure and create deformation into a 3 sphere. The rhombus is both different than a 3 sphere but also deformable into a sphere. This Poincare Conjecture can be both proven and disproven.
III.
ELLIPSE DISCUSSION
From Wikipedia.org an ellipse is “In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve.” [12] Some ellipses are here
[13]: Look at the third ellipse, which is not the same as a 3 sphere. Poincare Conjecture can be disproved by showing that manifolds like ellipses are not 3 spheres. Word meaning should mean something. An ellipse and a 3 sphere should be comprehended as having differences and different appearance! One could compress the shape and make it three dimensional however to create a 3 sphere. Growth or universal expansion of all the parts in the ellipse could produce or deform into a 3 sphere. We can argue for a Struck Flow like the Ricci Flow that an object can transform or deform into a 3 sphere. Struck Flow would not be like heat flow in Ricci’s work or Ricci’s Flow, but rather a transformation based on expansion or deflation so that one shape can be transformed into a 3-sphere shape. Poincare Conjecture can be proven also based on the ability to imagine one shape being transformed into a 3 sphere shape. Struck Flow does not involve heat, but rather expansion and deflation as deformation methods.
www.ijmrem.com
IJMREM
Page 46
Poincare Conjecture Proven and Disproved Using…
IV.
SQUARE DISCUSSION
From Wikipedia.org, a square is “In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles).[1] It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted ◻ {\displaystyle \square } ABCD.” [14] A square looks like
[14] A square is not a sphere. Since a square is not a 3 sphere, a square manifold is not homeomorphic with a 3 sphere. A square could be deformed into a sphere however through growth in various areas of the square structure. Still using the same definition of homeomorphic, a square is different than a 3 sphere. Poincare Conjecture disproved and Poincare conjecture supported using an analysis of a square.
V.
TRIANGLE DISCUSSION
A triangle is another basic shape in geometry. Here is how Wikipedia.org defines a triangle accessed on 2/5/2018 “Triangle From Wikipedia, the free encyclopedia This article is about the basic geometric shape. For other uses, see Triangle (disambiguation). Equilateral triangle
A regular triangle Type
Regular polygon
Edges and vertices Schläfli symbol
3 {3}
Coxeter diagram Symmetry group Internal angle (degrees)
60°
Dual polygon
Self
Properties
www.ijmrem.com
Dihedral (D3), order 2×3
Convex, cyclic, equilateral, isogonal, isotoxal
IJMREM
Page 47
Poincare Conjecture Proven and Disproved Using… Triangle
A triangle Edges and vertices Schläfli symbol
3 {3} (for equilateral) various methods; see below
Area Internal angle (degrees)
60° (for equilateral)
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted.”[16] So if we use the definition of homeomorphic meaning same, we see that triangles are not the same as 3 spheres. A triangle with 3 sides, 3 edges is not the same as a 3 sphere. If we use the definition of homeomorphic meaning deformable, we can do an expansion of the areas of the triangle into a 3 sphere. So we can both disprove the Poincare conjecture by showing that a triangle is not a 3 sphere and prove the Poincare conjecture showing that we can transform a triangle into a 3 sphere through deformation.
VI.
PARALLELOGRAM DISCUSSION
We can do the same type of discussion with a parallelogram. Here is what a parallelogram is from wikipeida.org accessed on 2/5/2018. “Parallelogram From Wikipedia, the free encyclopedia This article is about the quadrilateral shape. Parallelogram
This parallelogram is a rhomboid as it has no right angles and unequal sides. Type
www.ijmrem.com
quadrilateral
IJMREM
Page 48
Poincare Conjecture Proven and Disproved Using… Edges and vertices
4
Symmetry group C2, [2]+, (22) b × h (base × height); ab sin θ (product of adjacent sides and sine of any vertex angle)
Area Properties
convex
In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.” [17] So since we have parallel sides, we do not have a 3 sphere. Since a parallelogram has 4 sides, 4 sides are not the same as a 3 sphere. Poincare conjecture disproven as a quadrilateral shape with parallel 4 sides is not a 3 sphere shape with the edges all equal distance from the center as in a 3 sphere. On the other hand with the definition of homeomorphic as deformable, we could do a deformation of a parallelogram into a 3 sphere. Struck Flow could be used; Struck flow is expansion and growth and deflation or shrinking without involving heat or a Ricci Flow into a 3 sphere. Parallelograms are not the same as 3 spheres, but parallelograms can be deformed into 3 spheres. Poincare conjecture disproved and proven using the parallelogram case study.
VII.
CONCLUSIONS
Conjectures can be proved and disproved. Poincare Conjecture can easily be disproved as always changing forms or manifolds are not similar or deformable into 3 spheres. Poincare Conjecture can be easily proved also as a particular 3 sphere manifold is the same and deformable into a 3 sphere. Proof and disproof of the Poincare conjecture is possible. Discussion of the imaginary number, imaginary manifold, and always changing manifolds supports both proof and disproof of the Poincare Conjecture at the same time. Similarity and deformation both are possible and not possible between types of manifolds and 3 spheres. Suns and stars support also disproof and proof of the Poincare Conjecture. Linguistic discussion supports an argument that something always changing, imaginary, the Sun, the Stars are not really similar in form to stable 3 spheres. The idea of difference supports the idea that simply connected closed manifolds, or many diverse, varied forms, are not similar in form to 3 spheres as linguistically manifolds are different than 3 spheres. I find support for proof, disproof and ultimately disproof of the Poincare conjecture to maintain an idea of difference between manifolds and 3 spheres. Additionally, as we are not completely certain what Jules Henri Poincare meant by his conjecture; there is some concern as a type of disproof that we do not really know what Henri Poincare meant or was hypothesizing. Henri Poincare lived from 29 April 1854 – 17 July 1912, [18] we cannot directly speak with him about what his conjecture was discussing exactly. Henri Poincare invented his conjecture in about 1900 and 1904 and there is reason to believe he supported my disproof based on the Poincare Conjecture’s origin. See Wikipedia.org’s discussion of the Poincare conjecture accessed 12/31/2017 “Poincaré claimed in 1900 that homology, a tool he had devised based on prior work by Enrico Betti, was sufficient to tell if a 3-manifold was a 3-sphere. However, in a 1904 paper he described a counterexample to this claim, a space now called the Poincaré homology sphere.” [19] Further support that Jules Henri Poincare saw the Poincare homology sphere as a counterexample can be found.
“Poincaré
originally conjectured [4] that a homology 3-sphere (http://planetmath.org/HomologySphere) must be homeomorphic to S3. He later found a counterexample based on the group of rotations of the regular dodecahedron (http://planetmath.org/RegularPolyhedron), and restated his conjecture in of the fundamental group. (See [5]). To be accurate, the restatement took the form of a question. However it has always been referred to as Poincaré’s Conjecture.)” [20]
www.ijmrem.com
IJMREM
Page 49
Poincare Conjecture Proven and Disproved Using… The Poincare Homology sphere discussion shows that he knew of some support for a Poincare conjecture disproof very different than Grigori Perelman’s proof. Disproof can be supported by Poincare’s own Poincare homology sphere of 1904! My discussion of stars, Sun, imaginary numbers, imaginary manifolds, always changing manifolds more different than 3 spheres provides further support for a disproof of the Poincare conjecture. I submitted dozens of disproof’s and proofs also since 2010 from the disproof idea that I do not want to be deformed into a 3 sphere to the proof idea that a specific manifold of a 3 sphere is homeomorphic to a particular 3 sphere. An imaginary manifold existing only in imagination is NOT homeomorphic to a specific 3 sphere. There are mathematicians who object to the idea of imaginary numbers, imaginary numbers objected to can be seen more different than 3 spheres. “Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his La Géométrie, where the term imaginary was used and meant to be derogatory.[ [7][8]” [21] The rhombus and trapezoid discussion also shows that rhombus and trapezoids are different than 3 spheres, but also deformable into 3 spheres. Poincare Conjecture can be both disproven with the meaning of homeomorphic as same and proven with the meaning of homeomorphic as deformable. Ellipse provide more support that an ellipse is a manifold that is not the same as a 3 sphere, so Poincare Conjecture can be easily disproven. Ellipse are not equal to 3 spheres. When a star explodes or goes supernova, an exploding star is not similar to a 3 sphere. In addition, an ellipse can be transformed or flow into a 3 sphere so the Poincare Conjecture can also be supported or proven. Proof and disproof of the Poincare Conjecture are supported by examination of a number of cases and examples! Parallelograms, squares, triangles provide further support that these shapes are not homeomorphic or the same as 3 spheres, but these shapes can be deformed into 3 spheres. Poincare Conjecture is easily disproven given the meaning of homeomorphic as same and proven given the meaning of homeomorphic as deformable. The famous Poincare Conjecture can be disproven and proven using case studies of parallelograms, squares, rhombus, trapezoid, ellipse, imaginary numbers, imaginary manifold, stars, Suns, always changing manifolds. The original objection I had around July 2010 to the Poincare Conjecture that an always changing, many, diverse shape manifold is not the same as a 3 sphere remains. We can deform a manifold, or a shape always changing, diverse, varied shapes into a 3 sphere, but an always changing, diverse, varied manifold or changing or diverse shape is not the same as a 3 sphere. My objection that varied, diverse, changing shapes are not the same as 3 spheres applies as much to the work of Richard Hamilton and as much to Michael Freedman in 1982 as to Grigori Perelman’s work in 2006. Rene Descartes was known for his understanding of all life and philosophy linked to Geometry called Cartesian Philosophy. The Poincare Conjecture can be seen as an extension of the importance of Cartesian Philosophy. We can disagree with both Rene Descartes and Jules Henri Poincare and their assertions about geometry and topology. Something diverse and varied like a 2 dimensional or 1 dimensional shape or 10dimensional shape or 200-dimensional shape can be seen as not one to one to a ball in 4-dimensional space. A 3 Sphere is defined as “It consists of the set of points equidistant from a fixed central point in 4dimensional Euclidean space.” [22] A line, dodecahedron, parabola, hyperbola would not be one to one to the 3 spheres as the shapes are different. Let’s look at a set of points that are not equal distance from a fixed central point. The set of point’s not equal distance from the fixed central point is not going to be one to one to the 3 spheres or the set of points that is equal distance to a fixed set of points. I invent then the “Struck Manifold-Set of Points of all Different, Varied or Changing Distance from Fixed Central Point” to show that the Struck Manifold is not one to one to a set of points of equal distance to a central point! Manifolds of varied distance from central points are not homeomorphic or one to one to 3 spheres with all points equal distance to a central point. Acknowledgements- My mom Jane Frances Back Struck gave me the confidence to work on math problems by mom’s getting an A in Economics at Mundelein College and B in Psychology at Loyola University, so I wanted to thank her for her help. Mom died 7/15/2017. Thank you to Arthur Loevy, Jon Loevy and Andy Thayer who loaned me $50 to get the article published as journals pay fees to be indexed.
REFERENCE 1. 2. 3.
Webster’s New World Dictionary, p. 385. Pocket Books, New York. (1995). Weisstein, Eric W. "Homeomorphic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Homeomorphic.html Weinstein, Eric W. "Homeomorphic." From Math World--A Wolfram Web Resource.
www.ijmrem.com
IJMREM
Page 50
Poincare Conjecture Proven and Disproved Using‌ 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
https://www.merriam-webster.com/dictionary/manifold accessed on 12/31/2017 Weisstein, Eric W. "Imaginary Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ImaginaryNumber.html Palmer, Alex. Weird-o-pedia The Ultimate Book of Surprising, Strange, and Incredibly Bizarre Factors About (Supposedly) Ordinary Things. Skyhorse Publishing, New York, NY. (2012). https://en.wikipedia.org/wiki/Spicule/solar_physics accessed 4/4/2018. https://en.wikipedia.org/wiki/Milky_Way accessed 4/4/2018. https://en.wikipedia.org/wiki/Gregorio_Ricci-Curbastro accessed 12/31/2017. https://en.wikipedia.org/wiki/Trapezoid accessed 1/27/2018 https://en.wikipedia.org/wiki/Rhombus accessed 1/27/2018 https://en.wikipedia.org/wiki/Ellipse accessed 1/27/2018 https://en.wikipedia.org/wiki/Ellipse accessed 1/27/2018 https://en.wikipedia.org/wiki/Square accessed 1/29/2018 https://en.wikipedia.org/wiki/Square accessed 1/29/2018 https://en.wikipedia.org/wiki/Triangle accessed 2/5/2018 https://en.wikipedia.org/wiki/Parallelogram accessed 2/5/2018 https://en.wikipedia.org/wiki/Henri_Poincare accessed 4/5/2018 https://en.wikipedia.org/wiki/Poincareconjecture found 12/31/2017 http://planetmath.org/PoincareDodecahedralSpace accessed 4/5/2018 https://en.wikipedia.org/wiki/Imaginary_number accessed on 4/4/2018. https://en.wikipedia.org/wiki/3-sphere accessed on 4/5/2018
www.ijmrem.com
IJMREM
Page 51