Other problems are that the two remaining geometries are not necessarily infinite, they too could be finite. The first, Euclidian geometry, is where everything is flat and continues forever, and the angles in a triangle always add up to 180° no matter the size of the triangle, what we are used to dealling with. An example of a finite Euclidian surface would be a torus. The other geometry, hyperbolic, is in the shape a of saddle. In a hyperbolic plane the angles of a triangle add up to less than 180°, by how much depends on the size of the triangle. This could be finite in many ways making something like a torus with angles that must be less than the (number of sides minus two) times 180. An example would be an 18 sided polyhedra with each pair of opposite sides connected. Two lines of thought are that the Universe is "simply connected", that between two points there is exactly one straight line, or that the Universe is "multi connected" there are infinite straight lines between two points, as in a sphere.
A slight breeze blows across my face. It is summer, above the tree line. My hair gently flies back. There is a scent of new budding greenery. Closely hugging the ground, as if to attempt to cling on to the stones, are small bushes. Every now and then a bird flies from one to the other. Small birds stay close to the protection of bushes, in case a great bird of prey is above them. The birds always have this fear in the depths of the conscious, but it pales small in comparison to the spectacular strength of the weather.
When dark splotches can be seen creeping across the land, the rain distorts the view of the land, then it is time to hide. The clouds reach the summit and all becomes dark. Fog surrounds everything. The rain pelts against the rocks, making a rhythmic smacking sound and pushing the sturdy little bushes flat. Splashing in newly formed streams. These streams quickly grow, racing down the rock face, pulling with them all that lies in their path.
Then, the first sign of worse to come, the far off thud. Then again, only louder. It echoes and echoes, replaced by another but even more thunderous than the one before. As this great noise approaches, growing in volume and strength everything attempts to flee from the heights that only minutes before they loved. Now the birds fear these same treeless stones.
The flash lights the sky, warns every thing that the real danger has come. The fog that moments before obscured our vision now illuminates each drop of moisture that hangs in the air. At the same time a deafening crack splits through the air, raging through the mist, hitting the rock with the force able to almost knock anything over. Before the echo fades of another piercing spear of light illuminates the haze and an earth-shattering crash rocks the mountain. Again this happens, until I'm sure that fires have sprung to life in at least four places, but the pounding rain has put them out.
Only now does the storm start to move away. The lightning subsides and the thunder becomes just a faint drumming. The rain dies down to a light tapping on my skin before totally departing. Now with everything soaked to the core, there is finally calm. With fear and danger past beauty returns. The birds again start their tweeting and the brush is clearly seen.
The sun shines down on my back. The sun scorches everything beneath its glaring eye and the only shade is at the base of rocks. I can see all across the country side, over hills, across lakes, and all the way up the sides of other mountains. Towns look like little splotches of white against the expanse of green. Roads, like thin strips of gray with little sparkly dots, wind from one town to the next. What is the most beautiful from this height is the lakes. They are like sapphire gems displayed across an emerald background. When small clouds move across the sky I can see their shadows on the forest or towns. These shadow run across the land in a complex patter of lights and darks.
The wind pierces my jacket. It is fall above the tree line. The fierce vigor of the wind presses everything flat. The biting cold of the wind feels like a knifes edge brushing across the skin. As the wind twists and winds its way around the rocks it lets of a series of loud piercing howls. In a heart felt attempt to tear all thoughts from our minds. Despite the wind if one looks out across the land they see the most spectacular site, all the trees below seem to be aflame. Reds, oranges, and yellows are splayed across in a vibrant glowing glory of fall. Here below me is what should only occur in the lands of dreams. With out the wind there acting as an anchor to reality it would be easy to loose oneself in the beauty for endless ages.
Hurricane strength winds knock me to my knees. It is winter above tree line. The wind is a raging monster. Screaming louder than the thunder of the summer. With the power to catch an eight year old and send them flying off into the abyss of freezing air. It whips around the steel cables of a fire tower in attempts to rip it from the rock below. The tower itself bends against this awesome force leaning against its constraints of the wires. This all powerful wind carves the snow into huge compressed slabs. In places there are six feet of snow and ice. They end in jagged streaks jutting off into space, far enough that the unwary could step to close to the edge and the snow would break off leaving them helpless to the wind and fall. Even these dangerous outcrops are magnificently shaped into sharpened points, looking like shining white knives.
This is the mountain above tree line. In all its beauty it is horrific. With every jewel that the mountain holds there is a powerful force holding it there. The sights that can be seen are breathtaking and the storms that guard them are just as powerful. Autumn holds the wonders of the trees below, and winter the snow, ice and wind all cap the mountain above tree line.