Orlav asked, “What three dimensions familiar to you?”
Todd paused thoughtfully, “Well, two dimensions really describes the number of degrees east and west, and the number of degrees north and south. Of the equator, I suppose.”
“Example?”
Choosing one familiar to him, Todd replied, “Like, 'Where is the football?' Thirty yards west of the middle of the field, and twenty feet north of the southern sideline.
“Da, da”, Professor Orlav affirmed happily. “Do you know why to describe Space in two dimensions not work?”
Todd took a moment, reassembling his teacher’s words into familiar English, finally offering, “Well, for one thing Space isn't flat.”
Orlav continued, “Two dimensions only work if Space completely flat like you say. Like plate. Two-dimension: length by width.” He scanned the group again. “Questions?” No one raised a hand. He moved on. “OK, now we discuss three dimensions.” “Three dimensions helps – it adds depth, you see.” He scribbled notes on the board in near unintelligible prose.
“Imagine cross-country hiker in Switzerland plans trek of 100 kilometers to nearest town, but he forget two big mountains lie in-between. One route he can walk between mountains and other route he must climb over.” He drew a crude stick figure. “The hike between two mountains, or climb up and down for both, huge difference for time-line and supplies, da? Distance traveled here,” he tapped his finger on the climbing route, “much more than hiker walking on straight road,” this time drawing a flat line that went straight between the peaks.
“Can anyone think of another example of three-dimension travel?” There wasn’t a sound in the entire auditorium. A student near Todd tentatively raised her hand. Orlav didn’t know her name. “Yes, you there,” he called out, pointing to her. “Kak vas zavoot?”
She stared at him. He caught himself, “How do you call yourself?” he repeated carefully.
“Alyssa,” she replied hesitantly. “What example you have?”
“Air travel?
“Ochen khahrahshoh!” Orlav boomed happily. “There are other factors – wind, weather, load – but think how sad passengers would be if pilots only plan for distance in flat latitude and longitude.” The class ventured a cautious laugh. “What shape in geometry does jet really travel across country?” He nodded to Alyssa.
“It would have to be a kind of half-circle … a parabolic curve, I guess, to include the 'up' and the 'down'.” “Do you know how high jet stream is?”
“A couple of miles?” she guessed.
“Eleven kilometers,” Orlav emphasized, not clarifying much for the American students. “If jet short on fuel because they forget to calculate trip up and down from jet stream, result bad, nyet? Did you know, some of best jets in world are invented by Russian engineers?” The class smiled to themselves; he was renowned for his pride in his homeland.