If your eyes glazed over a bit there, you’re not alone. The picture and description are hard to parse. Before we try to understand the space itself, we should figure out why we would bother with it at all. Hatcher uses it as an example of a space that is “contractible but not in any obvious way.” We have therefore encouraged every provider to open their ‘online windows and doors’ to help you make an informed choice. The profile score is a measure of how much information has been provided.

Ican’t think about the house with two roomswithout hearing John 14:2: “In my father’s house there are many rooms.” This topological example is a bit more modest than the kingdom of God. After all, there are just two rooms.A more appropriate verse is this: In Bing’s house, there are two rooms. If it were not so, would Hatcher have told you how to construct it? EXPLORE THE WORLD OF BING: 4 different room sets to explore and enjoy. Including a fold-out kitchen with opening and closing cupboards.

Since 1995, Westwood Bing (a.k.a the five sisters) has actually been a scheduled monument and is now protected for posterity. They’ve become a symbol of West Lothian, and a memorial to the shale oil industry, yet the famous five-leafed bing is an oddity, a one of a kind. To build this space, start with a box divided into two chambers by a horizontal rectangle, where by a ‘rectangle’ we mean not just the four edges of a rectangle but also its interior. Access to the two chambers from outside the box is provided by two vertical tunnels. The upper tunnel is made by punching out a square from the top of the box and another square directly below it from the middle horizontal rectangle, then inserting four vertical rectangles, the walls of the tunnel. This tunnel allows entry to the lower chamber from outside the box. the lower tunnel is formed in similar fashion, providing entry to the upper chamber. Finally, two vertical rectangles are inserted to form ‘support walls’ for the two tunnels. The resulting space X thus consists of three horizontal pieces homeomorphic to annuli plus all the vertical rectangles that form the walls of the two chambers. If you want to take the easy way out instead, 12-year-old David Hitchman, son of University of Northern Iowa mathematician Theron J. Hitchman, made a fun video about building it in Minecraft, and Ken Baker wrote about it on the blog Sketches of Topology. There, he describes it in a different way:

I was skeptical of the house with two rooms for a long time. I just couldn’t see how to squish it up into a point without changing its topology. I took Hatcher’s word for it, but it wasn’t until I made it for myself that I truly believed in its contractibility. The day before spring break last semester, I brought some clay to my topology class, and one of the things we made was the house with two rooms. By the time we were finished, I felt like I truly understood the space. The house with two rooms...and a slightly leaky roof. ENCOURAGES IMAGINATIVE PLAY: 10 great accessories for imaginative play. Put Bing to bed or help him run a bath, lets use Brenda the Blender to bake a cake! David Schwimmer's 'Rachel' script slip up accidentally created one of the most awkward lines in FriendsThe house with two rooms is sometimes called Bing’s house after R. H. Bing, the mathematician who first described the space. But like many math students, I first encountered it in Allen Hatcher’s algebraic topology book. He gives us the following picture and definition of the space. Bing House Playset features popular CBeebies characters – Flop and Bing! Just like the house you see on the hit TV show. Beautifully made from tough, colourful plastic. Includes two posable figures and nine accessories.

Westwood bing looks very different because it was built by machines. All features of the Westwood works were very different to everything that had come before. Built to fuel Britain’s war machine the designers of Westwood works took full advantage of the technical advances that had taken place since the previous new oil works almost forty years before. This new technology extended to the disposal of the spent shale from the efficient new retorts. This finely graded waste was transported along a conveyor belt to a central bunker from where the spent shale was fed to a large carriage on rails. This was hauled to the top of the bing where the carriage was discharged by gravity.

You know how instead of doing a handshake, you could grab each other’s wrists? And your thumbs almost but don’t quite reach your fingers? Bing’s glove for two hands would fit snugly.