1 x xyz

M xy Adding fractions that have a common denominator : 3. M xy Adding fractions that have a common denominator : 6. Solving a Single Variable Equation : 8.

Why learn this. Terms and topics Adding subtracting finding least common multiple Reducing fractions to lowest terms. When the openings are closed up, the result is the Roman surface shown in Figure 3.

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A pair of lobes can be seen in the West and East directions of Figure 3. If the three intersecting hyperbolic paraboloids are drawn far enough that they intersect along the edges of a tetrahedron, then the result is as shown in Figure 4. One of the lobes is seen frontally—head on—in Figure 4. The lobe can be seen to be one of the four corners of the tetrahedron.

If the continuous surface in Figure 4 has its sharp edges rounded out—smoothed out—then the result is the Roman surface in Figure 5. One of the lobes of the Roman surface is seen frontally in Figure 5, and its bulbous — balloon-like—shape is evident. If the surface in Figure 5 is turned around degrees and then turned upside down, the result is as shown in Figure 6. Figure 6 shows three lobes seen sideways.

Between each pair of lobes there is a locus of double points corresponding to a coordinate axis. The three loci intersect at a triple point at the origin. The fourth lobe is hidden and points in the direction directly opposite from the viewer.

Multiplicative inverse

The Roman surface shown at the top of this article also has three lobes in sideways view. The Roman surface is non- orientablei.

This is not quite obvious. To see this, look again at Figure 3. Let this ant move North. As it moves, it will pass through the other two paraboloids, like a ghost passing through a wall. These other paraboloids only seem like obstacles due to the self-intersecting nature of the immersion.

Let the ant ignore all double and triple points and pass right through them. So the ant moves to the North and falls off the edge of the world, so to speak. It now finds itself on the northern lobe, hidden underneath the third paraboloid of Figure 3. The ant is standing upside-down, on the "outside" of the Roman surface. Let the ant move towards the Southwest.

It will climb a slope upside-down until it finds itself "inside" the Western lobe.

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As soon as the ant crosses this axis it will find itself "inside" the Northern lobe, standing right side up. Now let the ant walk towards the North. It will climb up the wall, then along the "roof" of the Northern lobe. The ant is back on the third hyperbolic paraboloid, but this time under it and standing upside-down.

Compare with Klein bottle. The Roman surface has four "lobes". Категория : Страницы значений по алфавиту. Пространства имён Статья Обсуждение. Просмотры Читать Править Править код История.