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%3 N_20210222203156c_ColinWright    By ColinWright 2021/02/22 @ 20:31:56c -------------------------------- We're in the world of Topology, so we can distort it.        (select only this node)     N_20210222203156d_ColinWright    By ColinWright 2021/02/22 @ 20:31:56d -------------------------------- Shrink the top edge and expand the bottom edge to get a lamp shade.        (select only this node)     N_20210222203156c_ColinWright->N_20210222203156d_ColinWright N_20210222193221b_ColinWright    By ColinWright 2021/02/22 @ 19:32:21b -------------------------------- Words sometimes have patterns and can be broken down and analysed        (select only this node)     N_20210222193221d_ColinWright    By ColinWright 2021/02/22 @ 19:32:21d -------------------------------- Prefixes: bio-, geo-, topo-        (select only this node)     N_20210222193221b_ColinWright->N_20210222193221d_ColinWright N_20210222194722a_ColinWright    By ColinWright 2021/02/22 @ 19:47:22a -------------------------------- This especially happens in medicine, where names of conditions, equipment, or treatments are often just literal Latin descriptions.        (select only this node)     N_20210222193221b_ColinWright->N_20210222194722a_ColinWright N_20210222193221c_ColinWright    By ColinWright 2021/02/22 @ 19:32:21c -------------------------------- Endings: -ology, -metry, -graphy        (select only this node)     N_20210222193221b_ColinWright->N_20210222193221c_ColinWright N_20210222223125a_ColinWright    By ColinWright 2021/02/22 @ 22:31:25a -------------------------------- Combining these gives us loads of words very quickly        (select only this node)     N_20210222193221c_ColinWright->N_20210222223125a_ColinWright N_20210222193221d_ColinWright->N_20210222223125a_ColinWright N_20210222193221e_ColinWright    By ColinWright 2021/02/22 @ 19:32:21e -------------------------------- "Topo" comes from the Greek: "Topos" meaning "place"        (select only this node)     N_20210222193221d_ColinWright->N_20210222193221e_ColinWright N_20210222193221h_ColinWright    By ColinWright 2021/02/22 @ 19:32:21h -------------------------------- Topography - the writing down of the shape of something        (select only this node)     N_20210222193221e_ColinWright->N_20210222193221h_ColinWright N_20210222193221f_ColinWright    By ColinWright 2021/02/22 @ 19:32:21f -------------------------------- Although we use it as "Shape"        (select only this node)     N_20210222193221e_ColinWright->N_20210222193221f_ColinWright N_20210222193221g_ColinWright    By ColinWright 2021/02/22 @ 19:32:21g -------------------------------- "Topometry" doesn't exist.        (select only this node)     N_20210222193221f_ColinWright->N_20210222193221g_ColinWright N_20210222193221f_ColinWright->N_20210222193221h_ColinWright N_20210222193221i_ColinWright    By ColinWright 2021/02/22 @ 19:32:21i -------------------------------- Topology - the study of shapes        (select only this node)     N_20210222193221f_ColinWright->N_20210222193221i_ColinWright N_20210222224033a_ColinWright    By ColinWright 2021/02/22 @ 22:40:33a -------------------------------- This stretching and deforming is why "Topometry" doesn't exist ... the deformations will change the distances and angles        (select only this node)     N_20210222193221g_ColinWright->N_20210222224033a_ColinWright N_20210222193958a_ColinWright    By ColinWright 2021/02/22 @ 19:39:58a -------------------------------- So consider a sheet of paper        (select only this node)     N_20210222193221i_ColinWright->N_20210222193958a_ColinWright N_20210222194115a_ColinWright    By ColinWright 2021/02/22 @ 19:41:15a -------------------------------- In the world of topology we don't care about exact distances or angles        (select only this node)     N_20210222193221i_ColinWright->N_20210222194115a_ColinWright N_20210222200905c_ColinWright    By ColinWright 2021/02/22 @ 20:09:05c -------------------------------- You now have a ... thing.        (select only this node)     N_20210222200905d_ColinWright    By ColinWright 2021/02/22 @ 20:09:05d -------------------------------- This is called a Moebius Band/Strip/Loop.        (select only this node)     N_20210222200905c_ColinWright->N_20210222200905d_ColinWright N_20210222203156e_ColinWright    By ColinWright 2021/02/22 @ 20:31:56e -------------------------------- Now squash it flat ... what do we have?        (select only this node)     N_20210222203156d_ColinWright->N_20210222203156e_ColinWright N_20210222193221a_ColinWright    By ColinWright 2021/02/22 @ 19:32:21a -------------------------------- A version of "Maths in a Twist"        (select only this node)     N_20210224205745a_ColinWright    By ColinWright 2021/02/24 @ 20:57:45a -------------------------------- This is a talk about Topology        (select only this node)     N_20210222193221a_ColinWright->N_20210224205745a_ColinWright N_20210222193221a_ColinWright->N_20210222193221b_ColinWright N_20210222203156f_ColinWright    By ColinWright 2021/02/22 @ 20:31:56f -------------------------------- A disk with a hole.        (select only this node)     N_20210222203156e_ColinWright->N_20210222203156f_ColinWright N_20210222193958b_ColinWright    By ColinWright 2021/02/22 @ 19:39:58b -------------------------------- Ask "How many sides?"        (select only this node)     N_20210222193958c_ColinWright    By ColinWright 2021/02/22 @ 19:39:58c -------------------------------- It's a "quadrilateral", which we can analyse as meaning "Four Sides"        (select only this node)     N_20210222193958b_ColinWright->N_20210222193958c_ColinWright N_20210222193958d_ColinWright    By ColinWright 2021/02/22 @ 19:39:58d -------------------------------- But when you write on a sheet of paper you can write on both sides        (select only this node)     N_20210222193958c_ColinWright->N_20210222193958d_ColinWright N_20210222194151a_ColinWright    By ColinWright 2021/02/22 @ 19:41:51a -------------------------------- This is what we do in the world of Geometry        (select only this node)     N_20210222193958c_ColinWright->N_20210222194151a_ColinWright N_20210222193958e_ColinWright    By ColinWright 2021/02/22 @ 19:39:58e -------------------------------- That implies that there are *two* sides - front and back        (select only this node)     N_20210222193958d_ColinWright->N_20210222193958e_ColinWright N_20210222193958f_ColinWright    By ColinWright 2021/02/22 @ 19:39:58f -------------------------------- That's the sense that we will use:        (select only this node)     N_20210222193958e_ColinWright->N_20210222193958f_ColinWright N_20210222193958g_ColinWright    By ColinWright 2021/02/22 @ 19:39:58g -------------------------------- A side is a surface you can paint.        (select only this node)     N_20210222193958f_ColinWright->N_20210222193958g_ColinWright N_20210222193958h_ColinWright    By ColinWright 2021/02/22 @ 19:39:58h -------------------------------- How many sides?        (select only this node)     N_20210222193958g_ColinWright->N_20210222193958h_ColinWright N_20210222204806b_ColinWright    By ColinWright 2021/02/22 @ 20:48:06b -------------------------------- A disk has two sides and one edge. Can we subtract a hole and get something with no edges?        (select only this node)     N_20210222204806c_ColinWright    By ColinWright 2021/02/22 @ 20:48:06c -------------------------------- Yes we can.        (select only this node)     N_20210222204806b_ColinWright->N_20210222204806c_ColinWright N_20210222193958i_ColinWright    By ColinWright 2021/02/22 @ 19:39:58i -------------------------------- How many different colours can you use if colours aren't allowed to meet on a side.        (select only this node)     N_20210222193958j_ColinWright    By ColinWright 2021/02/22 @ 19:39:58j -------------------------------- So a sheet of paper has two sides.        (select only this node)     N_20210222193958i_ColinWright->N_20210222193958j_ColinWright N_20210222195435a_ColinWright    By ColinWright 2021/02/22 @ 19:54:35a -------------------------------- So a disk has two sides and one edge.        (select only this node)     N_20210222193958j_ColinWright->N_20210222195435a_ColinWright N_20210222193958k_ColinWright    By ColinWright 2021/02/22 @ 19:39:58k -------------------------------- So what's this thing that forms the border between the two sides?        (select only this node)     N_20210222193958j_ColinWright->N_20210222193958k_ColinWright N_20210222193958l_ColinWright    By ColinWright 2021/02/22 @ 19:39:58l -------------------------------- We call that an "edge" ... just as a knife has an edge.        (select only this node)     N_20210222193958k_ColinWright->N_20210222193958l_ColinWright N_20210222193958m_ColinWright    By ColinWright 2021/02/22 @ 19:39:58m -------------------------------- Does the sheet have four edges?        (select only this node)     N_20210222193958l_ColinWright->N_20210222193958m_ColinWright N_20210222203156g_ColinWright    By ColinWright 2021/02/22 @ 20:31:56g -------------------------------- So a Cylinder is a Disk with a hole.        (select only this node)     N_20210222203156h_ColinWright    By ColinWright 2021/02/22 @ 20:31:56h -------------------------------- We can write that as C=D+h        (select only this node)     N_20210222203156g_ColinWright->N_20210222203156h_ColinWright N_20210222203156i_ColinWright    By ColinWright 2021/02/22 @ 20:31:56i -------------------------------- So going one step down, from D to C, is adding a hole        (select only this node)     N_20210222203156g_ColinWright->N_20210222203156i_ColinWright N_20210222193958n_ColinWright    By ColinWright 2021/02/22 @ 19:39:58n -------------------------------- We are in the world of topology, so we can distort surfaces        (select only this node)     N_20210222193958o_ColinWright    By ColinWright 2021/02/22 @ 19:39:58o -------------------------------- We can "smoosh" in a corner, or stretch out a side.        (select only this node)     N_20210222193958n_ColinWright->N_20210222193958o_ColinWright N_20210222193958p_ColinWright    By ColinWright 2021/02/22 @ 19:39:58p -------------------------------- So we can imagine stretching this out into a disk shape.        (select only this node)     N_20210222193958o_ColinWright->N_20210222193958p_ColinWright N_20210222193958q_ColinWright    By ColinWright 2021/02/22 @ 19:39:58q -------------------------------- And a disk will have just the one edge.        (select only this node)     N_20210222193958p_ColinWright->N_20210222193958q_ColinWright N_20210222193958q_ColinWright->N_20210222195435a_ColinWright N_20210222194115b_ColinWright    By ColinWright 2021/02/22 @ 19:41:15b -------------------------------- We are allowed to distort things, provided we don't tear them, punch holes in them, glue them, etc.        (select only this node)     N_20210222194115a_ColinWright->N_20210222194115b_ColinWright N_20210222194115a_ColinWright->N_20210222193221g_ColinWright N_20210222194115c_ColinWright    By ColinWright 2021/02/22 @ 19:41:15c -------------------------------- We can stretch or otherwise deform them.        (select only this node)     N_20210222194115b_ColinWright->N_20210222194115c_ColinWright N_20210222223507a_ColinWright    By ColinWright 2021/02/22 @ 22:35:07a -------------------------------- Imagine taking a sheet of paper ... it's a rectangle. Now trim a sliver off. To be absolutely precise it's now an irregular pentagon, but broadly we still think of it as being the same shape.        (select only this node)     N_20210222194115b_ColinWright->N_20210222223507a_ColinWright N_20210222194115c_ColinWright->N_20210222224033a_ColinWright N_20210222194115c_ColinWright->N_20210222193958n_ColinWright N_20210222200905g_ColinWright    By ColinWright 2021/02/22 @ 20:09:05g -------------------------------- Take your time.        (select only this node)     N_20210222204806d_ColinWright    By ColinWright 2021/02/22 @ 20:48:06d -------------------------------- Take a disk, and push down in the middle to make a shallow dish.        (select only this node)     N_20210222204806c_ColinWright->N_20210222204806d_ColinWright N_20210222193958h_ColinWright->N_20210222193958i_ColinWright N_20210222202103a_ColinWright    By ColinWright 2021/02/22 @ 20:21:03a -------------------------------- At this point it's worth starting to be a bit organised:        (select only this node)     N_20210222202103b_ColinWright    By ColinWright 2021/02/22 @ 20:21:03b -------------------------------- (Insert partially populated table)        (select only this node)     N_20210222202103a_ColinWright->N_20210222202103b_ColinWright N_20210222204806e_ColinWright    By ColinWright 2021/02/22 @ 20:48:06e -------------------------------- We're in the world of topology, so we're allowed to distort things        (select only this node)     N_20210222204806d_ColinWright->N_20210222204806e_ColinWright N_20210222193958a_ColinWright->N_20210222193958b_ColinWright N_20210222194722b_ColinWright    By ColinWright 2021/02/22 @ 19:47:22b -------------------------------- Hyper tachycardia        (select only this node)     N_20210222194722a_ColinWright->N_20210222194722b_ColinWright N_20210222194722c_ColinWright    By ColinWright 2021/02/22 @ 19:47:22c -------------------------------- Electrocardiogram (ECG)        (select only this node)     N_20210222194722a_ColinWright->N_20210222194722c_ColinWright N_20210222194722d_ColinWright    By ColinWright 2021/02/22 @ 19:47:22d -------------------------------- etc        (select only this node)     N_20210222194722b_ColinWright->N_20210222194722d_ColinWright N_20210222194722c_ColinWright->N_20210222194722d_ColinWright N_20210222195435b_ColinWright    By ColinWright 2021/02/22 @ 19:54:35b -------------------------------- Now from a sheet of A4 cut off the largest square you can, and the strip you get as an off-cut, cut that in half lengthwise.        (select only this node)     N_20210222195435a_ColinWright->N_20210222195435b_ColinWright N_20210222195927a_ColinWright    By ColinWright 2021/02/22 @ 19:59:27a -------------------------------- Counting carefully, a cylinder has two sides and two edges        (select only this node)     N_20210222195435a_ColinWright->N_20210222195927a_ColinWright N_20210222223704a_ColinWright    By ColinWright 2021/02/22 @ 22:37:04a -------------------------------- I'll keep asking this question        (select only this node)     N_20210222195435a_ColinWright->N_20210222223704a_ColinWright N_20210222195435a_ColinWright->N_20210222202103a_ColinWright N_20210222195435d_ColinWright    By ColinWright 2021/02/22 @ 19:54:35d -------------------------------- So now you have a square and two strips.        (select only this node)     N_20210222195435b_ColinWright->N_20210222195435d_ColinWright N_20210222195435c_ColinWright    By ColinWright 2021/02/22 @ 19:54:35c -------------------------------- (I'll give you a minute)        (select only this node)     N_20210222195435b_ColinWright->N_20210222195435c_ColinWright N_20210222204806f_ColinWright    By ColinWright 2021/02/22 @ 20:48:06f -------------------------------- Now lift the edge further to get a fruit bowl        (select only this node)     N_20210222204806e_ColinWright->N_20210222204806f_ColinWright N_20210222195435e_ColinWright    By ColinWright 2021/02/22 @ 19:54:35e -------------------------------- Bend one of the strips around and overlap the short sides        (select only this node)     N_20210222195435d_ColinWright->N_20210222195435e_ColinWright N_20210222200905a_ColinWright    By ColinWright 2021/02/22 @ 20:09:05a -------------------------------- Take one of your strips, bend it round, and before you overlap the short sides, put a half twist.        (select only this node)     N_20210222195435d_ColinWright->N_20210222200905a_ColinWright N_20210222195435f_ColinWright    By ColinWright 2021/02/22 @ 19:54:35f -------------------------------- (Notice I'm using words from geometry to describe things)        (select only this node)     N_20210222195435e_ColinWright->N_20210222195435f_ColinWright N_20210222195435g_ColinWright    By ColinWright 2021/02/22 @ 19:54:35g -------------------------------- We get a cylinder        (select only this node)     N_20210222195435e_ColinWright->N_20210222195435g_ColinWright N_20210222204806g_ColinWright    By ColinWright 2021/02/22 @ 20:48:06g -------------------------------- Start to shrink the edge over to make a fishbowl        (select only this node)     N_20210222204806f_ColinWright->N_20210222204806g_ColinWright N_20210222195435h_ColinWright    By ColinWright 2021/02/22 @ 19:54:35h -------------------------------- (In "normal" geometry a cylinder is a solid object, but in topology it's this thing)        (select only this node)     N_20210222195435i_ColinWright    By ColinWright 2021/02/22 @ 19:54:35i -------------------------------- So the question: How many sides and how many edges?        (select only this node)     N_20210222195435i_ColinWright->N_20210222195927a_ColinWright N_20210222204806h_ColinWright    By ColinWright 2021/02/22 @ 20:48:06h -------------------------------- Shrink the edge even further, and what do you have?        (select only this node)     N_20210222204806g_ColinWright->N_20210222204806h_ColinWright N_20210222200905b_ColinWright    By ColinWright 2021/02/22 @ 20:09:05b -------------------------------- (This can take some time)        (select only this node)     N_20210224220402a_ColinWright    By ColinWright 2021/02/24 @ 22:04:02a -------------------------------- We have a Disk, a Cylinder, and a Moebius.        (select only this node)     N_20210222202103b_ColinWright->N_20210224220402a_ColinWright N_20210222203156a_ColinWright    By ColinWright 2021/02/22 @ 20:31:56a -------------------------------- How do we populate this table? Can we find something with two sides and 3 edges?        (select only this node)     N_20210222202103b_ColinWright->N_20210222203156a_ColinWright N_20210222204806i_ColinWright    By ColinWright 2021/02/22 @ 20:48:06i -------------------------------- (Take a moment)        (select only this node)     N_20210222204806h_ColinWright->N_20210222204806i_ColinWright N_20210222204806j_ColinWright    By ColinWright 2021/02/22 @ 20:48:06j -------------------------------- You now have a sphere with a hole.        (select only this node)     N_20210222204806h_ColinWright->N_20210222204806j_ColinWright N_20210222195435g_ColinWright->N_20210222195435i_ColinWright N_20210222195435g_ColinWright->N_20210222195435h_ColinWright N_20210222203731c_ColinWright    By ColinWright 2021/02/22 @ 20:37:31c -------------------------------- Insert diagram of a disk with two holes        (select only this node)     N_20210222203731d_ColinWright    By ColinWright 2021/02/22 @ 20:37:31d -------------------------------- So we can see that every time we punch a hole in something we get an extra edge.        (select only this node)     N_20210222203731c_ColinWright->N_20210222203731d_ColinWright N_20210222193958m_ColinWright->N_20210222193958n_ColinWright N_20210222201415a_ColinWright    By ColinWright 2021/02/22 @ 20:14:15a -------------------------------- We find that a Moebius Strip has one side and one edge.        (select only this node)     N_20210222195927a_ColinWright->N_20210222201415a_ColinWright N_20210222195927a_ColinWright->N_20210222202103a_ColinWright N_20210222195927a_ColinWright->N_20210222200905a_ColinWright N_20210222200905a_ColinWright->N_20210222200905b_ColinWright N_20210222200905a_ColinWright->N_20210222200905c_ColinWright N_20210222203156b_ColinWright    By ColinWright 2021/02/22 @ 20:31:56b -------------------------------- Let's take another look at a cylinder        (select only this node)     N_20210222203156a_ColinWright->N_20210222203156b_ColinWright N_20210222210025a_ColinWright    By ColinWright 2021/02/22 @ 21:00:25a -------------------------------- So we can fill in the hole, and we get a sphere, which has two sides and no edges        (select only this node)     N_20210222204806j_ColinWright->N_20210222210025a_ColinWright N_20210222204806k_ColinWright    By ColinWright 2021/02/22 @ 20:48:06k -------------------------------- So a disk is, in the world of topology, the same as a sphere with a hole.        (select only this node)     N_20210222204806j_ColinWright->N_20210222204806k_ColinWright N_20210222200905e_ColinWright    By ColinWright 2021/02/22 @ 20:09:05e -------------------------------- So, what questions am I going to ask?        (select only this node)     N_20210222200905d_ColinWright->N_20210222200905e_ColinWright N_20210222200905f_ColinWright    By ColinWright 2021/02/22 @ 20:09:05f -------------------------------- How many sides? How many edges?        (select only this node)     N_20210222200905e_ColinWright->N_20210222200905f_ColinWright N_20210224220722a_ColinWright    By ColinWright 2021/02/24 @ 22:07:22a -------------------------------- So that's the warm-up ... now for something completely different.        (select only this node)     N_20210222201415a_ColinWright->N_20210224220722a_ColinWright N_20210222201415a_ColinWright->N_20210222202103a_ColinWright N_20210222203156b_ColinWright->N_20210222203156c_ColinWright N_20210222204806l_ColinWright    By ColinWright 2021/02/22 @ 20:48:06l -------------------------------- But also C=D+h, and now we have D=S+h.        (select only this node)     N_20210222204806k_ColinWright->N_20210222204806l_ColinWright N_20210222200905f_ColinWright->N_20210222200905g_ColinWright N_20210222200905f_ColinWright->N_20210222201415a_ColinWright N_20210222203156j_ColinWright    By ColinWright 2021/02/22 @ 20:31:56j -------------------------------- What might come below the disk?        (select only this node)     N_20210222203156i_ColinWright->N_20210222203156j_ColinWright N_20210222203731a_ColinWright    By ColinWright 2021/02/22 @ 20:37:31a -------------------------------- If stepping down the table from D to C is adding a hole, maybe stepping down from C is adding another hole        (select only this node)     N_20210222203156j_ColinWright->N_20210222203731a_ColinWright N_20210222205723a_ColinWright    By ColinWright 2021/02/22 @ 20:57:23a -------------------------------- That also shows how we can get something with one side and two edges        (select only this node)     N_20210222203731d_ColinWright->N_20210222205723a_ColinWright N_20210222203731e_ColinWright    By ColinWright 2021/02/22 @ 20:37:31e -------------------------------- So going down the table is adding holes, what will it be going up the table?        (select only this node)     N_20210222203731d_ColinWright->N_20210222203731e_ColinWright N_20210222204806m_ColinWright    By ColinWright 2021/02/22 @ 20:48:06m -------------------------------- So C=S+2h ... take a sphere and punch a hole in the top and bottom ... now you have a cylinder!        (select only this node)     N_20210222204806l_ColinWright->N_20210222204806m_ColinWright N_20210222203156f_ColinWright->N_20210222203156g_ColinWright N_20210222204806a_ColinWright    By ColinWright 2021/02/22 @ 20:48:06a -------------------------------- We can fill in a hole, which we can think of as "taking a hole away" or "subtracting a hole".        (select only this node)     N_20210222203731e_ColinWright->N_20210222204806a_ColinWright N_20210222203156h_ColinWright->N_20210222203156j_ColinWright N_20210222203731b_ColinWright    By ColinWright 2021/02/22 @ 20:37:31b -------------------------------- So under C is C+h, which is D+2h.        (select only this node)     N_20210222203731a_ColinWright->N_20210222203731b_ColinWright N_20210222203731b_ColinWright->N_20210222203731c_ColinWright N_20210222214447a_ColinWright    By ColinWright 2021/02/22 @ 21:44:47a -------------------------------- What about "going up" from a Moebius Strip? Can we "subtract a hole" from a Moebius Strip?        (select only this node)     N_20210222204806a_ColinWright->N_20210222214447a_ColinWright N_20210222204806a_ColinWright->N_20210222204806b_ColinWright N_20210222205723b_ColinWright    By ColinWright 2021/02/22 @ 20:57:23b -------------------------------- Punch a hole in a Moebius Strip!        (select only this node)     N_20210222205723a_ColinWright->N_20210222205723b_ColinWright N_20210224221055a_ColinWright    By ColinWright 2021/02/24 @ 22:10:55a -------------------------------- Use the square you have left over, and create a big plus sign.        (select only this node)     N_20210224221752a_ColinWright    By ColinWright 2021/02/24 @ 22:17:52a -------------------------------- Use two opposite arms, bend them around, and make a Moebius Strip.        (select only this node)     N_20210224221055a_ColinWright->N_20210224221752a_ColinWright N_20210224222936a_ColinWright    By ColinWright 2021/02/24 @ 22:29:36a -------------------------------- Someone once asked Roger Federer what was so good about Switzerland. His answer was ...        (select only this node)     N_20210224221055a_ColinWright->N_20210224222936a_ColinWright N_20210224221055b_ColinWright    By ColinWright 2021/02/24 @ 22:10:55b -------------------------------- Use two opposite arms to make a cylinder.        (select only this node)     N_20210224221055a_ColinWright->N_20210224221055b_ColinWright N_20210222205723c_ColinWright    By ColinWright 2021/02/22 @ 20:57:23c -------------------------------- So we can complete our table going down.        (select only this node)     N_20210222214447f_ColinWright    By ColinWright 2021/02/22 @ 21:44:47f -------------------------------- Let's take a cylinder and cut it in half around the equator. What do we get?        (select only this node)     N_20210222205723c_ColinWright->N_20210222214447f_ColinWright N_20210222205723c_ColinWright->N_20210222214447a_ColinWright N_20210222210025b_ColinWright    By ColinWright 2021/02/22 @ 21:00:25b -------------------------------- It's not the only thing with two sides and no edges ... another is a "Torus" ... the surface of a ring doughnut, or an old-fashioned car inner-tube.        (select only this node)     N_20210222210025a_ColinWright->N_20210222210025b_ColinWright N_20210222210025c_ColinWright    By ColinWright 2021/02/22 @ 21:00:25c -------------------------------- Or a Double Torus, or Triple Torus, etc. They all have two sides and no edges.        (select only this node)     N_20210222210025b_ColinWright->N_20210222210025c_ColinWright N_20210222210025d_ColinWright    By ColinWright 2021/02/22 @ 21:00:25d -------------------------------- So just because two things match in numbers of sides and edges, that doesn't necessarily mean they are the same thing.        (select only this node)     N_20210222210025c_ColinWright->N_20210222210025d_ColinWright N_20210222210025e_ColinWright    By ColinWright 2021/02/22 @ 21:00:25e -------------------------------- But if they differ in the number of sides or the number of edges (or both) then they cannot be the same thing.        (select only this node)     N_20210222210025d_ColinWright->N_20210222210025e_ColinWright N_20210222210025f_ColinWright    By ColinWright 2021/02/22 @ 21:00:25f -------------------------------- This is one way to determine that two things are definitely different.        (select only this node)     N_20210222210025e_ColinWright->N_20210222210025f_ColinWright N_20210222210025f_ColinWright->N_20210224220722a_ColinWright N_20210222214447b_ColinWright    By ColinWright 2021/02/22 @ 21:44:47b -------------------------------- A Moebius Strip has one edge, so we can try to imagine that as being the edge of a hole, and patch it by gluing a disk onto it.        (select only this node)     N_20210222214447a_ColinWright->N_20210222214447b_ColinWright N_20210222221255a_ColinWright    By ColinWright 2021/02/22 @ 22:12:55a -------------------------------- We don't get a Klein Bottle        (select only this node)     N_20210222214447b_ColinWright->N_20210222221255a_ColinWright N_20210222214447d_ColinWright    By ColinWright 2021/02/22 @ 21:44:47d -------------------------------- The thing we get is called the Real Projective Plane, or RP^2, and it's a wonderfully fascinating thing that would take another two hours to investigate.        (select only this node)     N_20210222214447b_ColinWright->N_20210222214447d_ColinWright N_20210222214447c_ColinWright    By ColinWright 2021/02/22 @ 21:44:47c -------------------------------- Possible conceptually, but not "for real" in our 3D space.  It must self-intersect.        (select only this node)     N_20210222214447b_ColinWright->N_20210222214447c_ColinWright N_20210224221349a_ColinWright    By ColinWright 2021/02/24 @ 22:13:49a -------------------------------- Bend the other two arms *away* from the cylinder, and make a Moebius Strip.        (select only this node)     N_20210224221055b_ColinWright->N_20210224221349a_ColinWright N_20210224221055c_ColinWright    By ColinWright 2021/02/24 @ 22:10:55c -------------------------------- Bend the other two arms *away* from the cylinder, and make another cylinder.        (select only this node)     N_20210224221055b_ColinWright->N_20210224221055c_ColinWright N_20210222214447e_ColinWright    By ColinWright 2021/02/22 @ 21:44:47e -------------------------------- We won't do that now.        (select only this node)     N_20210222214447d_ColinWright->N_20210222214447e_ColinWright N_20210222221437a_ColinWright    By ColinWright 2021/02/22 @ 22:14:37a -------------------------------- Just as the sphere and the torus are different things, both with two sides and no edges, so the Klein Bottle and the Projective Plane are different things (for each other), both with one side and no edges.        (select only this node)     N_20210222214447d_ColinWright->N_20210222221437a_ColinWright N_20210224221055d_ColinWright    By ColinWright 2021/02/24 @ 22:10:55d -------------------------------- How many sides? How many edges?        (select only this node)     N_20210224221055c_ColinWright->N_20210224221055d_ColinWright N_20210222214447g_ColinWright    By ColinWright 2021/02/22 @ 21:44:47g -------------------------------- Not hard to visualise, we get two cylinders of half the height.        (select only this node)     N_20210222214447f_ColinWright->N_20210222214447g_ColinWright N_20210222214447h_ColinWright    By ColinWright 2021/02/22 @ 21:44:47h -------------------------------- So let's take our Moebius Strip and cut that in half around the centre line        (select only this node)     N_20210222214447g_ColinWright->N_20210222214447h_ColinWright N_20210222214447i_ColinWright    By ColinWright 2021/02/22 @ 21:44:47i -------------------------------- What have we got?        (select only this node)     N_20210222214447h_ColinWright->N_20210222214447i_ColinWright N_20210222214447j_ColinWright    By ColinWright 2021/02/22 @ 21:44:47j -------------------------------- How many sides? How many edges?        (select only this node)     N_20210222214447i_ColinWright->N_20210222214447j_ColinWright N_20210222214447i_ColinWright->N_20210224220722a_ColinWright N_20210224221238a_ColinWright    By ColinWright 2021/02/24 @ 22:12:38a -------------------------------- It has two sides and one edge, but it's clearly not a disk.        (select only this node)     N_20210224221238b_ColinWright    By ColinWright 2021/02/24 @ 22:12:38b -------------------------------- What do you get when you cut it in half        (select only this node)     N_20210224221238a_ColinWright->N_20210224221238b_ColinWright N_20210222221949b_ColinWright    By ColinWright 2021/02/22 @ 22:19:49b -------------------------------- But why are they linked?        (select only this node)     N_20210222221949c_ColinWright    By ColinWright 2021/02/22 @ 22:19:49c -------------------------------- That comes back to the question of twistiness, and whether we care.        (select only this node)     N_20210222221949b_ColinWright->N_20210222221949c_ColinWright N_20210224221238c_ColinWright    By ColinWright 2021/02/24 @ 22:12:38c -------------------------------- Cut all the way around both "equators", and across the crossing point.        (select only this node)     N_20210224221238b_ColinWright->N_20210224221238c_ColinWright N_20210222223507b_ColinWright    By ColinWright 2021/02/22 @ 22:35:07b -------------------------------- This is what topology tries to capture ... you're allowed to distort things and still call them the "same shape"        (select only this node)     N_20210222223507b_ColinWright->N_20210222194115c_ColinWright N_20210224221055d_ColinWright->N_20210224221238a_ColinWright N_20210222205723b_ColinWright->N_20210222205723c_ColinWright N_20210224221238d_ColinWright    By ColinWright 2021/02/24 @ 22:12:38d -------------------------------- Could you have predicted that?        (select only this node)     N_20210224221238c_ColinWright->N_20210224221238d_ColinWright N_20210222215213a_ColinWright    By ColinWright 2021/02/22 @ 21:52:13a -------------------------------- We end up with two sides and two edges ... is it a cylinder?        (select only this node)     N_20210222215730a_ColinWright    By ColinWright 2021/02/22 @ 21:57:30a -------------------------------- Could we have predicted this result?        (select only this node)     N_20210222215213a_ColinWright->N_20210222215730a_ColinWright N_20210222215213b_ColinWright    By ColinWright 2021/02/22 @ 21:52:13b -------------------------------- There are two schools of thought:        (select only this node)     N_20210222215213a_ColinWright->N_20210222215213b_ColinWright N_20210222215213d_ColinWright    By ColinWright 2021/02/22 @ 21:52:13d -------------------------------- No, it's not a cylinder, because the edges "wind around" each other.        (select only this node)     N_20210222215213b_ColinWright->N_20210222215213d_ColinWright N_20210222215213c_ColinWright    By ColinWright 2021/02/22 @ 21:52:13c -------------------------------- Yes, it's a cylinder, because if you were an ant and living on the surface you wouldn't be able to tell that it has twists.        (select only this node)     N_20210222215213b_ColinWright->N_20210222215213c_ColinWright N_20210222215323a_ColinWright    By ColinWright 2021/02/22 @ 21:53:23a -------------------------------- So it depends on whether you care about the twistiness of things.        (select only this node)     N_20210222215213c_ColinWright->N_20210222215323a_ColinWright N_20210222215213d_ColinWright->N_20210222215323a_ColinWright N_20210222215323c_ColinWright    By ColinWright 2021/02/22 @ 21:53:23c -------------------------------- In the world of computer networks and data analysis, we don't.        (select only this node)     N_20210222215323a_ColinWright->N_20210222215323c_ColinWright N_20210222215323b_ColinWright    By ColinWright 2021/02/22 @ 21:53:23b -------------------------------- In the world of things like protein folding or movements of robot arms, we do care        (select only this node)     N_20210222215323a_ColinWright->N_20210222215323b_ColinWright N_20210222215334a_ColinWright    By ColinWright 2021/02/22 @ 21:53:34a -------------------------------- So as I say, it depends.        (select only this node)     N_20210222215323b_ColinWright->N_20210222215334a_ColinWright N_20210222215323c_ColinWright->N_20210222215334a_ColinWright N_20210222223507a_ColinWright->N_20210222223507b_ColinWright N_20210222214447j_ColinWright->N_20210222215213a_ColinWright N_20210222214447j_ColinWright->N_20210222215730a_ColinWright N_20210222223704a_ColinWright->N_20210222200905e_ColinWright N_20210224221522a_ColinWright    By ColinWright 2021/02/24 @ 22:15:22a -------------------------------- How many sides? How many edges?        (select only this node)     N_20210224221349a_ColinWright->N_20210224221522a_ColinWright N_20210222215334a_ColinWright->N_20210222221949c_ColinWright N_20210224221752b_ColinWright    By ColinWright 2021/02/24 @ 22:17:52b -------------------------------- Bend the other two arms *away* from it, and make another Moebius Strip.        (select only this node)     N_20210224221752c_ColinWright    By ColinWright 2021/02/24 @ 22:17:52c -------------------------------- How many sides? How many edges?        (select only this node)     N_20210224221752b_ColinWright->N_20210224221752c_ColinWright N_20210222215730b_ColinWright    By ColinWright 2021/02/22 @ 21:57:30b -------------------------------- Yes, we have a way of drawing diagrams that let us analyse the results of operations like this.        (select only this node)     N_20210222215730c_ColinWright    By ColinWright 2021/02/22 @ 21:57:30c -------------------------------- It's easy when things are small and simple, like these, but as things get more complicated we need formal ways of deciding what we want to do.        (select only this node)     N_20210222215730b_ColinWright->N_20210222215730c_ColinWright N_20210222221830c_ColinWright    By ColinWright 2021/02/22 @ 22:18:30c -------------------------------- Draw the diagram for the Moebius Strip and mark where it will be cut.        (select only this node)     N_20210222215730b_ColinWright->N_20210222221830c_ColinWright N_20210222215730d_ColinWright    By ColinWright 2021/02/22 @ 21:57:30d -------------------------------- So we can devise something, then analyse whether we get the result we want, and if not, do something different.        (select only this node)     N_20210222215730c_ColinWright->N_20210222215730d_ColinWright N_20210222220725a_ColinWright    By ColinWright 2021/02/22 @ 22:07:25a -------------------------------- What if we cut a Moebius Strip in thirds?        (select only this node)     N_20210222215730d_ColinWright->N_20210222220725a_ColinWright N_20210222220450a_ColinWright    By ColinWright 2021/02/22 @ 22:04:50a -------------------------------- One thing to note is that the Moebius Strip has one edge. When we cut it down the middle, maybe that one edge will hold the whole thing together.        (select only this node)     N_20210222220450a_ColinWright->N_20210222220725a_ColinWright N_20210222220725b_ColinWright    By ColinWright 2021/02/22 @ 22:07:25b -------------------------------- Before making it, draw two lines down the strip of paper, dividing it into three equal sections.        (select only this node)     N_20210222220725a_ColinWright->N_20210222220725b_ColinWright N_20210222220725c_ColinWright    By ColinWright 2021/02/22 @ 22:07:25c -------------------------------- Now make the Moebius Strip        (select only this node)     N_20210222220725b_ColinWright->N_20210222220725c_ColinWright N_20210224221522b_ColinWright    By ColinWright 2021/02/24 @ 22:15:22b -------------------------------- It has one side and one edge, but it's clearly not a Moebius Strip.        (select only this node)     N_20210224221522c_ColinWright    By ColinWright 2021/02/24 @ 22:15:22c -------------------------------- Can you predict what you will get when you cut it in half?        (select only this node)     N_20210224221522b_ColinWright->N_20210224221522c_ColinWright N_20210222220725d_ColinWright    By ColinWright 2021/02/22 @ 22:07:25d -------------------------------- Before cutting it, can you predict what you will get?        (select only this node)     N_20210222220725e_ColinWright    By ColinWright 2021/02/22 @ 22:07:25e -------------------------------- Now cut along the lines ... what do you get?        (select only this node)     N_20210222221830a_ColinWright    By ColinWright 2021/02/22 @ 22:18:30a -------------------------------- Could you have predicted that?        (select only this node)     N_20210222220725e_ColinWright->N_20210222221830a_ColinWright N_20210222224033a_ColinWright->N_20210222193958a_ColinWright N_20210224221522d_ColinWright    By ColinWright 2021/02/24 @ 22:15:22d -------------------------------- Try it ...        (select only this node)     N_20210224221522c_ColinWright->N_20210224221522d_ColinWright N_20210224221752a_ColinWright->N_20210224221752b_ColinWright N_20210222221437b_ColinWright    By ColinWright 2021/02/22 @ 22:14:37b -------------------------------- And just as there is a Double Torus, Triple Torus, etc, so there are other things with one side and no edges.        (select only this node)     N_20210222221437a_ColinWright->N_20210222221437b_ColinWright N_20210222221547a_ColinWright    By ColinWright 2021/02/22 @ 22:15:47a -------------------------------- There is a way to classify these things, and to know that we have included everything.        (select only this node)     N_20210222221437b_ColinWright->N_20210222221547a_ColinWright N_20210224215959a_ColinWright    By ColinWright 2021/02/24 @ 21:59:59a -------------------------------- In particular, every 2D surface is a sphere with some number of holes. Some of those holes have Moebius Strips patched into them, some are connected by cylinders, and some are left as holes.        (select only this node)     N_20210222221547a_ColinWright->N_20210224215959a_ColinWright N_20210222221547b_ColinWright    By ColinWright 2021/02/22 @ 22:15:47b -------------------------------- This is a bit like a Periodic Table of Surfaces.        (select only this node)     N_20210222221547a_ColinWright->N_20210222221547b_ColinWright N_20210222221547c_ColinWright    By ColinWright 2021/02/22 @ 22:15:47c -------------------------------- Once we know that we have a whole number of protons in the nucleus of an atom, we know for sure that we haven't missed any elements.        (select only this node)     N_20210222221547b_ColinWright->N_20210222221547c_ColinWright N_20210224205745b_ColinWright    By ColinWright 2021/02/24 @ 20:57:45b -------------------------------- What is Topology?        (select only this node)     N_20210224205745a_ColinWright->N_20210224205745b_ColinWright N_20210222221830d_ColinWright    By ColinWright 2021/02/22 @ 22:18:30d -------------------------------- Make the central section really small and what's left will be cutting a Moebius Strip in half        (select only this node)     N_20210222221830c_ColinWright->N_20210222221830d_ColinWright N_20210222221830e_ColinWright    By ColinWright 2021/02/22 @ 22:18:30e -------------------------------- Make the central section really wide and what you get it just the entire Moebius Strip.        (select only this node)     N_20210222221830c_ColinWright->N_20210222221830e_ColinWright N_20210222221949a_ColinWright    By ColinWright 2021/02/22 @ 22:19:49a -------------------------------- So we must end up with two loops, one is a twisted cylinder, the other is a Moebius Strip.        (select only this node)     N_20210222221830d_ColinWright->N_20210222221949a_ColinWright N_20210222221830e_ColinWright->N_20210222221949a_ColinWright N_20210222221949a_ColinWright->N_20210222221949b_ColinWright N_20210224221522a_ColinWright->N_20210224221522b_ColinWright N_20210222221949c_ColinWright->N_20210224220722a_ColinWright N_20210224210719a_ColinWright    By ColinWright 2021/02/24 @ 21:07:19a -------------------------------- We will come back to this.        (select only this node)     N_20210222221949c_ColinWright->N_20210224210719a_ColinWright N_20210224220402a_ColinWright->N_20210222203156a_ColinWright N_20210222221255a_ColinWright->N_20210222221437a_ColinWright N_20210224220722a_ColinWright->N_20210224221055a_ColinWright N_20210222215730a_ColinWright->N_20210222215730b_ColinWright N_20210222215730a_ColinWright->N_20210222220450a_ColinWright N_20210224205745b_ColinWright->N_20210222193221i_ColinWright N_20210224221752d_ColinWright    By ColinWright 2021/02/24 @ 22:17:52d -------------------------------- We now have one side, but two edges ... is it a Moebius Strip with a hole in it? Or something else?        (select only this node)     N_20210224221752c_ColinWright->N_20210224221752d_ColinWright N_20210222221830a_ColinWright->N_20210224220722a_ColinWright N_20210222221830b_ColinWright    By ColinWright 2021/02/22 @ 22:18:30b -------------------------------- Short answer, yes.        (select only this node)     N_20210222221830a_ColinWright->N_20210222221830b_ColinWright N_20210224222508a_ColinWright    By ColinWright 2021/02/24 @ 22:25:08a -------------------------------- This comes back to the question of whether we care about links and twistiness.        (select only this node)     N_20210224210719a_ColinWright->N_20210224222508a_ColinWright N_20210224221943a_ColinWright    By ColinWright 2021/02/24 @ 22:19:43a -------------------------------- Can you predict what you will get when you cut it in half?        (select only this node)     N_20210224221752d_ColinWright->N_20210224221943a_ColinWright N_20210222221830b_ColinWright->N_20210222221830c_ColinWright N_20210224215959b_ColinWright    By ColinWright 2021/02/24 @ 21:59:59b -------------------------------- And that's it!        (select only this node)     N_20210224215959a_ColinWright->N_20210224215959b_ColinWright N_20210224215959a_ColinWright->N_20210222221547b_ColinWright N_20210224221943b_ColinWright    By ColinWright 2021/02/24 @ 22:19:43b -------------------------------- Try it ... does the result surprise you?        (select only this node)     N_20210224221943a_ColinWright->N_20210224221943b_ColinWright N_20210222220725c_ColinWright->N_20210222220725d_ColinWright N_20210222220725c_ColinWright->N_20210222220725e_ColinWright N_20210224221943c_ColinWright    By ColinWright 2021/02/24 @ 22:19:43c -------------------------------- Why are there two different results?        (select only this node)     N_20210224221943b_ColinWright->N_20210224221943c_ColinWright N_20210224221943d_ColinWright    By ColinWright 2021/02/24 @ 22:19:43d -------------------------------- The linkedness depends on whether you twisted the Moebius Strips in the same direction, or opposite directions.        (select only this node)     N_20210224221943c_ColinWright->N_20210224221943d_ColinWright N_20210224221943d_ColinWright->N_20210224222508a_ColinWright N_20210224222508b_ColinWright    By ColinWright 2021/02/24 @ 22:25:08b -------------------------------- We can apply these ideas in the movements of robot arms, and then it really does matter whether something gets tied in knots or not.        (select only this node)     N_20210224222508a_ColinWright->N_20210224222508b_ColinWright N_20210224222508c_ColinWright    By ColinWright 2021/02/24 @ 22:25:08c -------------------------------- We can apply these ideas in connectivity of communications systems, and then we don't care if things go over or under.        (select only this node)     N_20210224222508a_ColinWright->N_20210224222508c_ColinWright N_20210224222541a_ColinWright    By ColinWright 2021/02/24 @ 22:25:41a -------------------------------- So it's very much a case of: "It depends"        (select only this node)     N_20210224222508b_ColinWright->N_20210224222541a_ColinWright N_20210224222508c_ColinWright->N_20210224222541a_ColinWright N_20210224222936b_ColinWright    By ColinWright 2021/02/24 @ 22:29:36b -------------------------------- "Well, the flag is a big plus."        (select only this node)     N_20210224222936a_ColinWright->N_20210224222936b_ColinWright N_20210224225909a_ColinWright    By ColinWright 2021/02/24 @ 22:59:09a -------------------------------- Start        (select only this node)     N_20210224225909a_ColinWright->N_20210222193221a_ColinWright