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Hex reasoned that a moving point traces out a line segment, and that a line segment moving parallel to itself makes a square. Continuing with her logic, Hex realized that a square moving parallel to itself would make a "super-square", a 3-D cube. So, a cube in dimension N can be swept out by an N-1 dimensional cube (sometimes called an N-1 dimensional "face") moving "parallel to itself". Let's think about how many N-1 dimensional faces are on the boundary of an N-cube. For example, if we move a point (0-cube), we sweep out a line, which has 2 points on its boundary. If we move a line parallel to itself, we generate a square, which has 4 lines on its boundary (i.e., the square has four 1-D faces). A moving square can sweep out a 3-cube, which has 6 square 2-D faces. If a 3-D cube sweeps out a 4-D hypercube, how many 3-D faces are on the boundary of the 4-D hypercube? .. A moving point sweeps out a line A moving line sweeps out a square.

Hex reasoned that a moving point traces out a line segment, and that a line segment moving parallel to itself makes a square. Continuing with her logic, Hex realized that a square moving parallel to itself would make a "super-square", a 3-D cube. So, a cube in dimension N can be swept out by an N-1 dimensional cube (sometimes called an N-1 dimensional "face") moving "parallel to itself". Let's think about how many N-1 dimensional faces are on the boundary of an N-cube. For example, if we move a point (0-cube), we sweep out a line, which has 2 points on its boundary. If we move a line parallel to itself, we generate a square, which has 4 lines on its boundary (i.e., the square has four 1-D faces). A moving square can sweep out a 3-cube, which has 6 square 2-D faces. If a 3-D cube sweeps out a 4-D hypercube, how many 3-D faces are on the boundary of the 4-D hypercube? .. A moving point sweeps out a line A moving line sweeps out a square.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 16EQ
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Hex reasoned that a moving point traces out a line segment, and that a line segment moving parallel to itself makes a square. Continuing with her logic, Hex realized that a square moving parallel to itself would make a "super-square", a 3-D cube. So, a cube in dimension N can be swept out by an N-1 dimensional cube (sometimes called an N-1 dimensional "face") moving "parallel to itself". Let's think about how many N-1 dimensional faces are on the boundary of an N-cube. For example, if we move a point (0-cube), we sweep out a line, which has 2 points on its boundary. If we move a line parallel to itself, we generate a square, which has 4 lines on its boundary (i.e., the square has four 1-D faces). A moving square can sweep out a 3-cube, which has 6 square 2-D faces. If a 3-D cube sweeps out a 4-D hypercube, how many 3-D faces are on the boundary of the 4-D hypercube? .. A moving point sweeps out a line A moving line sweeps out a square.
Transcribed Image Text:Hex reasoned that a moving point traces out a line segment, and that a line segment moving parallel to itself makes a square. Continuing with her logic, Hex realized that a square moving parallel to itself would make a "super-square", a 3-D cube. So, a cube in dimension N can be swept out by an N-1 dimensional cube (sometimes called an N-1 dimensional "face") moving "parallel to itself". Let's think about how many N-1 dimensional faces are on the boundary of an N-cube. For example, if we move a point (0-cube), we sweep out a line, which has 2 points on its boundary. If we move a line parallel to itself, we generate a square, which has 4 lines on its boundary (i.e., the square has four 1-D faces). A moving square can sweep out a 3-cube, which has 6 square 2-D faces. If a 3-D cube sweeps out a 4-D hypercube, how many 3-D faces are on the boundary of the 4-D hypercube? .. A moving point sweeps out a line A moving line sweeps out a square.
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