answersLogoWhite

0


Best Answer

You have to think 3D!!! A triangular based pyramid does the trick if all the edges are of equal length.

More generally, you can arrange n+1 points equidistantly (Euclidean distance) in an n dimensional space, but it is simple to do it in an n+1 dimensional space. First an alogrithm for the n+1 dimensional space followed by an algorithm for an n dimensional space: For both algorithms: Define e_i as a unit vector in the direction of the ith axis for 1<= i <= n+1.

#1: n+1 dimensions

Place the ith point at e_i.

You are done since the distance between any two points is just sqrt(2).

#2: n dimensions

Place the first point at the origin

Place the second point at e_1.

For the (k+1)th point (with k= 2 through n) iterate the following procedure:

1) Let X_k be the center of mass of the first k points (this will be a point only involving the first k-1 coordinates).

2) Locate the next point at X_k + sqrt(1- X_k ^2)*e_k. This means the first k-1 coordinates are the coordinates of the center of mass, X_k. The kth coordinate is computed by first taking the sum of the squares of the coordinates of X_k. Subtracting this value from 1 then taking the square root yields the value of the kth coordinate.

When you have located the n+1 point, you are done. The proof of the equidistance is simple by induction. Assume that the first r points are equidistant (which is obvious for r=2 since a set two points must be equidistant). Then the distance that separates the first r points from each other must be 1 because that is the distance between the first two. Now consider the r+1 point. It's distance from all other points must be equal because all other points are symmetric about the center of mass and it sits directly "above" the center of mass in one higher dimension. We chose its distance above the center of mass to ensure that its distance from the origin is exactly 1 so its distance from each of the first r points is exactly 1. Since we know that the first r points are all exactly 1 apart from each other, we have showed that the first r+1 points are equidistant. This completes the induction proof.

User Avatar

Wiki User

16y ago
This answer is:
User Avatar

Add your answer:

Earn +20 pts
Q: How to arrange four points at equidistance?
Write your answer...
Submit
Still have questions?
magnify glass
imp
Related questions

What is the point from which all points on the circumference of a circle are equidistance?

The centre


What is an equidistance?

Equidistance refers to an equal distance between two points or objects. It can be used in various contexts, such as map making or mathematics, to illustrate equal spacing or balance.


Can you arrange the numbers 1-10 on the points of intersection so that the four numbers on each line total 22?

On a five point star


How could you arrange four points so that each of them was exactly the same distance from the other three?

use a ruler and measure it to see how far apart they should be.


How do you place four points equidistant from each other?

To place four points equidistant from each other, you would need to arrange them in the shape of a perfect square. This means that each point would be the same distance away from the other three points, forming equal sides of the square. The distance between each point can be calculated using the Pythagorean theorem if the coordinates of the points are known.


A set of points arrange in a row?

They could be the coordinates of a straight line equation


What are the four stages of the briefing process?

There are several steps that make up a 'construct a briefing' stage. These include collecting material, selecting visual aids, arranging key points, and establishing wording.


How can you arrange the letters elsactniehbov to get a four letter word?

can this be love?


Arrange the four numbers to form proportion that is true?

40,50,80,64


How many ways can you arrange four letters?

24 ways.


How many ways can you arrange four people on a bench?

24


What is the definition of parallel lines?

Two straight lines that are equidistance from each other.