[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-40807-en":3,"doc-seo-40807-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},40807,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Stephen Hawking Space and Time Warps","A lecture explores how space and time warps—common tropes in science fiction—relate to real physics and evolving geometry. Starting from Euclidean geometry’s assumption that triangle angles sum to 180 degrees, it contrasts flat space with curved, non-Euclidean geometry using the Earth’s great circles and spherical triangles. It argues that higher-dimensional embedding and large-scale curvature would change observable geometry, illustrated by an analogy of painters layering paint on a large ball.","Space and Time Warps  \nThis lecture is the intellectual property of Professor S.W. Hawking. You may not reproduce, editor distribute this document in anyway for monetary advantage.  \nIn science fiction, space and time warps are a commonplace. They are used for rapid journeys around the galaxy, or for travel through time. But today's science fiction, is often tomorrow's science fact. So what are the chances for space and time warps.  \nThe idea that space and time can be curved, or warped, is fairly recent. For more than two thousand years, the axioms of Euclidean geometry, were considered to be self evident. As those of you that were forced to learn Euclidean geometry at school may remember, one of the consequences of these axioms is, that the angles of a triangle, add up to a hundred and 80 degrees.  \nHowever, in the last century, people began to realize that other forms of geometry were possible, in which the angles of a triangle, need not add up to a hundred and 80 degrees. Consider, for example, the surface of the Earth. The nearest thing to a straight line on the surface of the Earth, is what is called, a great circle. These are the shortest paths between two points, so they are the roots that air lines use . Consider now the triangle on the surface of the Earth, made up of the equator, the line of 0 degrees longitude through London, and the line of 90 degrees longtitude east, through Bangladesh. The two lines of longitude, meet the equator at a right angle, 90 degrees. The two lines of longitude also meet each other at the north pole, at aright angle, or 90 degrees. Thus one has a triangle with three right angles. The angles of this triangle add up to two hundred and seventy degrees. This is greater than the hundred and eighty degrees, for a triangle on a flat surface. If one drew a triangle on a saddle shaped surface, one would find that the angles added up to less than a hundred and eighty degrees. The surface of the Earth, is what is called a two dimensional space. That is, you can move on the surface of the Earth, in two directions at right angles to each other: you can move north south, or east west. But of course, there is a third direction at right angles to these two, and that is up or down. That is to say, the surface of the Earth exists in three-dimensional space. The three dimensional space is flat. That is to say, it obeys Euclidean geometry. The angles of a triangle, add up to a hundred and eighty degrees. However, one could imagine a race of two dimensional creatures, who could move about on the surface of the Earth, but who couldn't experience the third direction, of up or down. They wouldn't know about the flat three-dimensional space, in which the surface of the Earth lives. For them, space would be curved, and geometry would be non-Euclidean.  \nIt would be very difficult to design a living being that could exist in only two dimensions.  \nFood that the creature couldn't digest would have to be spat out the same way it came in. If there were a passage right the way through, like we have, the poor animal would fall apart.  \nSo three dimensions, seems to be the minimum for life. But just as one can think of two dimensional beings living on the surface of the Earth, so one could imagine that the three dimensional space in which we live, was the surface of a sphere, in another dimension that we don't see . If the sphere were very large, space would be nearly flat, and Euclidean geometry would be a very good approximation over small distances. But we would notice that Euclidean geometry broke down, over large distances. As an  \nillustration of this, imagine a team of painters, adding paint to the surface of a large ball. Asthe thickness of the paint layer increased, the surface area would go up. If the ball were in a flat three-dimensional space, one could go on adding paint indefinitely, and the ball would get bigger and bigger. However, if the three-dimensional space, were really the surface of a sphere in an","cbCaia3eYguvpHGL","https://ap.wps.com/l/cbCaia3eYguvpHGL","pdf",136192,3,1,6,"English","en",105,"# Space and time warps in science fiction\n## From Euclidean to non-Euclidean geometry\n## Great circles and spherical triangles\n## Implications for life and observable curvature\n## Paint layers on a large ball","[{\"question\":\"What does the lecture say about space and time warps in science fiction versus real science?\",\"answer\":\"It treats warp ideas as familiar in fiction, while emphasizing that today’s science fiction can become tomorrow’s scientific fact. It then asks what the actual chances are for such phenomena.\"},{\"question\":\"How does the lecture use Earth to explain non-Euclidean geometry?\",\"answer\":\"It explains that great circles provide the shortest paths on Earth’s surface and uses spherical triangles to show that triangle angles can sum to 270 degrees rather than 180. This demonstrates how curvature changes geometry.\"},{\"question\":\"Why can’t a two-dimensional creature fully understand our geometry?\",\"answer\":\"The lecture argues that if beings could only move on a 2D surface, they would experience curved space and non-Euclidean rules. They would be unable to perceive the third direction (up/down) that makes the underlying 3D space flat.\"}]",1783315261,15,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"stephen-hawking-space-and-time-warps","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/stephen-hawking-space-and-time-warps/40807/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-12","2026-07-06",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the lecture say about space and time warps in science fiction versus real science?","Question",{"text":75,"@type":76},"It treats warp ideas as familiar in fiction, while emphasizing that today’s science fiction can become tomorrow’s scientific fact. It then asks what the actual chances are for such phenomena.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the lecture use Earth to explain non-Euclidean geometry?",{"text":80,"@type":76},"It explains that great circles provide the shortest paths on Earth’s surface and uses spherical triangles to show that triangle angles can sum to 270 degrees rather than 180. This demonstrates how curvature changes geometry.",{"name":82,"@type":73,"acceptedAnswer":83},"Why can’t a two-dimensional creature fully understand our geometry?",{"text":84,"@type":76},"The lecture argues that if beings could only move on a 2D surface, they would experience curved space and non-Euclidean rules. 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