Jump to content
BrainDen.com - Brain Teasers
  • 0

3D Connect the Dots



A spider is going to weave its new territory attaching to all the tips of wine corks 
at the cellar that are forming the 8 corners of a perfect cube (1 cu.m). The spider
must conserve its web fluid (with least linear meter used) and  move from cork to cork.
Can you describe the shape of its web? How long it is?

Link to comment
Share on other sites

11 answers to this question

Recommended Posts

  • 0

It appears the shortest overall length of spinneret thread would be ≈ 6.196152 meters.
The red lines in Plainglazed figure would each be √3/3 meters in length, the green lines would each be √3/6 meters in length, and the central yellow line would be (1 - √3/3) meters in length. This should total (3 + 9√3)/3 meters.

Link to comment
Share on other sites

  • 0

Connecting the 8 corks with web with least linear meter requires connecting with the web along the sides of the cube (no diagonals) with length 1m, and uses at least 7 segments to connect 8 corks. so the total length is 7*1m = 7 meters. and the shape could vary, as long as you pass the web only along the sides of the cube.

  • Upvote 1
Link to comment
Share on other sites

  • 0

Using the strategy I used in the first connect the dots, this is my answer.


Connect 4 corks with a spread out X. All 4 sides of the X are spread outward to make a 0.5 meter line connecting the two sides. Change the diagonal sides so that they also go up 0.25 meters. Do the same for the top half and connect them with a 0.5 meter line.

The diagonal sides are equal to sqrt(0.375). Multiply that by 8 for all 8 of those lines, and add the connecting 1.5 meters. This is equal to about 6.399 meters.


Link to comment
Share on other sites

  • 0



adding Bb3k's headless stick figure and crunching some numbers:


8(2x2+.52).5+ 4(1-2x)/3.5+ 1-2x-(1-2x)/3.5

f(x) = 8(2x2+1/4).5 + 3(1-2x)/3.5 + 1–2x

f’(x) = 16x/(2x2+1/4).5 – 2(3.5) – 2

f’(x) = 0 when x = ~.195

f(x=.195) = 4.568 + 1.409 + .258 = 6.235


the sqrt(3) comes from TSLF's 2D connect the dots:


f(x) = 4(x2+.52).5 + 1 – 2x

f”(x) = 4x/(x2+1/4).5 – 2

f’(x) = 0 when x = 1/2(3.5) = ~.289

tan-1(1/2(3.5)/.5) or tan-1(1/3.5) = 30°





Link to comment
Share on other sites

  • 0

Spoiler for some supporting math


I have the same calculations as DejMar but with an assumption that I cannot yet prove yields a minimum web length.  In my last response I thought this problem must have some grand symmetry and assumed the red lines in my drawing (above) must be the same distance x from their nearest two sides of the cube described by the eight corks. The following assumes a magical 30 degree angle of those red lines:


f(x) = 8/sqrt(3) + 4(1-2x)/sqrt(3) + 1-2sqrt(1/12-x2)-(1-2x)/sqrt(3)
f(x) = 8/sqrt(3) + 3(1-2x)/sqrt(3) + 1-2sqrt(1/12-x2)
f(x) = 8/sqrt(3) + sqrt(3)(1-2x) + 1-2sqrt(1/12-x2)
f'(x) = -2sqrt(3) + 2x/sqrt(1/12-x2)
f'(x) = 0 when x = .25

f(x=.25) = 1+9/sqrt(3) ≈ 6.196 


Link to comment
Share on other sites

  • 0

Alright, you got it.. I don't know if the spider can do that . Maybe with its  farthest pair of eyes.
All webs made a 120 degrees angle with the adjacent web like the towns & roads problem.

You can always minimize piping or wiring lengths using the same principle. The only task is
how to find the points to make 120deg network to connect N dots.

Again, good solving to all .. thanks.

Link to comment
Share on other sites

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

Answer this question...

×   Pasted as rich text.   Paste as plain text instead

  Only 75 emoji are allowed.

×   Your link has been automatically embedded.   Display as a link instead

×   Your previous content has been restored.   Clear editor

×   You cannot paste images directly. Upload or insert images from URL.


  • Recently Browsing   0 members

    • No registered users viewing this page.
  • Create New...