Question

The closed unit ball in \(\mathbb{R}^{3}\) centered at the origin is the set \(\left\\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 1\right\\} .\) Describe the following alternative unit balls. a. \(\\{(x, y, z):|x|+|y|+|z| \leq 1\\}.\) b. \(\\{(x, y, z): \max \\{|x|,|y|,|z|\\} \leq 1\\},\) where \(\max \\{a, b, c\\}\) is the maximum value of \(a, b,\) and \(c.\)

Short answer

Question: Describe the shapes of the alternative unit balls given by the inequalities (a) \(|x| + |y| + |z| \leq 1\) and (b) \(\max\{|x|, |y|, |z|\} \leq 1\). Answer: The alternative unit ball (a) has the shape of an octahedron, while the alternative unit ball (b) has the shape of a cube.

Step-by-step solution

Description of Unit Ball (a)

We are given the unit ball described by the inequality \(|x| + |y| + |z| \leq 1\). This inequality can be interpreted geometrically as the sum of the absolute values of the coordinates being less than or equal to 1. Since the absolute values take into account the "length" of the coordinate, we can think of this ball as an "octahedron" shape, with each coordinate axis forming one of its edges. To visualize this shape, imagine a pyramid made of four triangular faces on each octant of the \(\mathbb{R}^3\) space, meeting at the origin and the tips of the pyramids touching at the points \((\pm 1, 0, 0)\), \((0, \pm 1, 0)\), and \((0, 0, \pm 1)\). Therefore, the alternative unit ball (a) has the shape of an octahedron.

Description of Unit Ball (b)

Now let's analyze the alternative unit ball (b), given by the inequality \(\max\{|x|, |y|, |z|\} \leq 1\). Geometrically, this inequality means that the maximum of the absolute values of the coordinates must be less than or equal to 1. To visualize this shape, let's consider the case when the absolute value of each coordinate is equal to 1: $$|x| = |y| = |z| = 1$$ In this case, we can see that the largest "cuboid" (with its sides parallel to the coordinate axes) that would fit these conditions would be a cube centered at the origin, with its vertices at the points \((\pm 1, \pm 1, \pm 1)\). Therefore, the alternative unit ball (b) has the shape of a cube.

Key concepts

Geometric Shapes

In mathematics, geometric shapes are the forms we observe in two or three dimensions. They are fundamental to geometry, and each shape has distinct properties and characteristics.

Geometric shapes can be simple, like lines and circles, or more complex, like polyhedra and spheres. When dealing with shapes in three-dimensional space, such as unit balls, it's important to understand the arrangement of points that make up these structures.

• Lines: Defined by two points, extending infinitely in both directions.
• Planes: Flat surfaces that extend infinitely in two dimensions.
• Solid shapes: Include objects like cubes, spheres, and pyramids.

The unit ball serves as an example of a geometric concept, where it represents the set of points within a certain distance (or radius) from a central point, in this case, the origin of the coordinate system. It's a way of visualizing and understanding the constraints set by mathematical equations in a three-dimensional space.

Octahedron

An octahedron is a polyhedron with eight triangular faces, twelve edges, and six vertices. It's a type of solid shape and is one of the classical "Platonic solids." These shapes are highly symmetrical and have identical faces.

In the context of the given exercise, the unit ball \((x, y, z): |x| + |y| + |z| \leq 1\) forms an octahedron when plotted in a three-dimensional space. The reason it forms an octahedron is due to the nature of the inequality: the sum of the absolute values of coordinates remains within a boundary.

• Vertices: Points like \((\pm 1, 0, 0)\) are the vertices of the octahedron formed.
• Edges: They are formed by the lines between these points, aligned with the axes.
• Symmetry: It has axial symmetry, meaning it looks the same along any of its axes.

By visualizing this shape, students can better grasp how each coordinate is constrained, and how those constraints contribute to forming a geometrical shape like the octahedron.

Cube

A cube is a solid shape with six equal square faces, twelve edges, and eight vertices. It is a very familiar shape in geometric terms and is often used as a basic example of a three-dimensional shape.

In the exercise, the alternative unit ball \((x, y, z): \max \{|x|, |y|, |z|\} \leq 1\) is interpreted as a cube. This comes from the definition of the inequality which requires each coordinate's absolute value not to exceed 1, effectively forming the shape of a cube when plotted.

• Vertices: These occur at the extreme points, \((\pm 1, \pm 1, \pm 1)\).
• Faces: Each face is a square, and there are six such faces on a cube.
• Symmetry: The cube is symmetrical along its axes, making its analysis easier.

The cube in this context helps to illustrate how boundaries or limits applied to coordinates shape can confine them into a well-known geometrical figure like this. Understanding the cube in three dimensions aids in easier computation and visualization of certain mathematical problems.

Inequalities in Geometry

Inequalities in geometry are used to describe relationships and constraints between different components of geometric figures. They are essential tools for defining regions and boundaries within geometrical spaces.

In this exercise, inequalities help form unit balls in \(R^3\) by constraining the values of the coordinates. The two specific inequalities provided result in the creation of distinct geometric shapes:

• Absolute Value Inequality: \(|x| + |y| + |z| \leq 1\) forms an octahedron, emphasizing how collective distances combine to create boundaries.
• Maximal Value Inequality: \(\max \{|x|, |y|, |z|\} \leq 1\) leads to a cube, focusing on the furthest reach of any individual coordinate.

These inequalities allow for a clear depiction of the space within defined limits. Understanding them is crucial for mastering geometric problems and for practical applications in fields like architecture and physics where space and boundaries are pivotal.