As my mother is equally afraid of small spaces as she is of large spaces, I will build my mother a geodesic dome, its outer wall comprised of two walls, essentially a dome over a dome, three feet of space between the two domes. The dome is then a very large space, but comprised of many small spaces. Because a geodesic dome is made up of many triangles and because—if the two domes are three feet apart—we may put narrow landings in every triangle, so that the inhabitant, or visitor, might sit, rest or hide in any one of the many triangles.
I am about six feet tall, but, bent in half, I fit in any triangle with three sides of three feet (area: 3.9 feet).
To make the geodesic dome, rows of slightly smaller triangles are arranged upon slightly larger triangles which are arranged below slightly smaller triangles, and on and on until the pattern forms a dome (this is oversimplified of course; in reality, the larger triangles form pentagons and the smaller triangles form hexagons and the dome is formed by rows of pentagons followed by rows of hexagons, and so on; but this does not change the fact that we’re looking at triangles).
If my mother is able to reach up to feel with her hands something up to seven feet tall, then my mother will be able to reach up and feel around from any bottommost triangle to any second bottommost triangle. If I provide her with a stick, three feet in length, she will be able to poke around in any third level triangle.
Now, if it took my mother ten seconds to, in total darkness, determine whether or not I was inside any given triangle, and, if my mother went about this systematically (not wildly running triangle to triangle wherever her fear compels her, screaming and calling for me) checking triangle after triangle, it would take her thirty seconds to check each column, bottom- most triangle to topmost triangle.
My mother could then, within reason–at ten seconds per triangle, three triangles per column–examine two columns per minute, or, one-hundred and twenty columns an hour. If, given that this is the Pacific Northwest and at this time of year we have roughly eight hours every night of total darkness, I believe my mother could (again if she is rational and efficient) examine nine-hundred and sixty triangles per night.
- I will never hide in any triangle which my mother cannot reach.
- I will never switch triangles at any point in the night. When darkness falls and I enter the dome, I pick one triangle and I stay in it.
- On any given night, I may enter the dome from any direction and climb into any triangle, so long as I don’t violate rules one or two.
On any given night then, my mother’s odds of finding me are dependent totally on the size of the dome. Given a dome with the diameter of one hundred feet and the height of fifty feet, and therefore with the dome surface area of 15,707.96 feet, we divide 15,707.96 feet by the area of one triangle, so 15,707.96 feet divided by 3.9 feet, equalling 4027.68, rounding up to 4028 total triangles.
Given that my mother is only capable of searching the first three triangles in any given column, and given the diminishing number of triangles (a loss of six per hexagon every row moving up–the largest number at the bottom of the dome and the smallest number, being the dome’s topmost point, one hexagon at the top), row-by-row, a dome with 4028 triangles, as my mother would only be capable of searching roughly ten percent (402 triangles) of the surface area of a dome fifty feet high (at 120 triangles examined per hour), it would take my mother less than half a night to examine every triangle in all three bottommost rows of the dome. Inevitably, she would find me.
Now, if we doubled the height and diameter of the dome (so, d=200 feet, h=100 feet) to get a dome surface area of 62,831.85, we then get 16,110.73 or, rounding up, 16,111 triangles. Searchability is diminished to seven percent because the dome is higher. My mother then is capable of searching 805 triangles. Again, piece of cake for my mother.
But if I design a dome one-thousand feet in diameter and five-hundred feet in height (dome surface area: 1,570,796.33 feet) then my mother must contend with 402,768 total triangles, five thousand searchable triangles, her chances of finding me in one night are reduced to roughly one night in five nights.
And a dome of one-hundred thousand feet in diameter and fifty-thousand feet in height (dome surface area: 15,707,963,267.95 feet)? My mother has an estimated 5,034,604 searchable triangles. A dome of 5,034,604 searchable triangles has a total volume of 261,799,387,799,149.4 cubic feet, the same volume as if Lake Michigan flooded Lake Huron.
In a dome of this size, my mother, on any given night, has a 1 in 5594 chance in finding me. If she spends every night systematically examining triangles (maintaining her rate of 960 triangles examined per night), feeling around in every triangle in the dark, and poking at the upper triangles with a stick, I will see my mother only once every fifteen years.
Dr. Maev Barba attended the Puget Sound Writer’s Conference in 2018. She is a PNW native and a great lover of books. She used to sell books door-todoor. A doctor of astronomy, Barba looks into space and considers neither the small as too little, nor the large as too great, for the lover of stars knows there is no limit to dimension.