{"id":28,"date":"2026-01-25T00:27:10","date_gmt":"2026-01-25T00:27:10","guid":{"rendered":"https:\/\/learnx.ca\/um\/?page_id=28"},"modified":"2026-02-16T21:18:17","modified_gmt":"2026-02-16T21:18:17","slug":"geometry","status":"publish","type":"page","link":"https:\/\/imaginethis.ca\/u\/geometry\/","title":{"rendered":"Geometry"},"content":{"rendered":"\n

“One geometry cannot be more true than another; it can only be more convenient.”
\u2015Henri Poincare<\/p>\n\n\n\n

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Parallel lines can meet?<\/h2>\n\n\n\n
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A riddle<\/strong><\/h2>\n\n\n\n

Pose this riddle to your students:<\/p>\n\n\n\n

Molly steps out of her tent. She walks 1 km south and 1 km west. She sees a bear and gets scared.
She runs 1 km north\u2014and arrives back at her tent.
How is this possible? And what colour is the bear?<\/strong><\/p>\n\n\n\n

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If you\u2019re not sure of the answer, you\u2019re not alone!<\/p>\n\n\n\n

To explore the surprising geometry behind this riddle\u2014and what it reveals about parallel lines on curved surfaces<\/strong>\u2014watch this interview with Dr. Megumi Harada<\/strong> from McMaster University<\/em>. at imaginethis.ca\/megumi-harada<\/a><\/p>\n\n\n\n

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Parallel Lines in History<\/strong><\/h2>\n\n\n\n

About 2,300 years ago<\/strong>, Euclid attempted to prove his Parallel Lines Postulate<\/em>\u2014but he never succeeded. Neither did the many mathematicians who tried after him.
It turns out that the statement \u201cparallel lines never meet\u201d<\/strong> is not<\/em> a theorem that can be proven from simpler ideas. Instead, it is an assumption<\/strong>. And when you change that assumption, you get entirely different geometries.<\/p>\n\n\n

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Here is the question at the heart of Euclid\u2019s struggle:<\/p>\n\n\n\n

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Given a straight line and a point not on that line, how many straight lines can you draw through the point that never intersect the original line?<\/strong><\/p>\n<\/blockquote>\n\n\n\n

Euclid believed the answer must be one<\/strong>, but he could not prove it.<\/p>\n\n\n\n

Different answers lead to different geometric worlds:<\/p>\n\n\n

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