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Can anyone explain why comet lander Philae has only three legs? Isn't it obvious that it now has problems that it would not have had if it had four legs arranged as a regular tetrahedron - or even more legs? Were legs so expensive?

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  • $\begingroup$ have a look What geometry concept can be used to explain this? "The reason actually lies in the fact that it only takes three points to define a plane. Any point that is added to that plane will make it harder and harder for the plane to be stable." geom.uiuc.edu/~wanous/part3.html $\endgroup$ – anna v Nov 13 '14 at 19:49
  • $\begingroup$ We are not talking about something like a plane underground with strong gravitational effects - but about an unpredictabily irregular underground with some soft gravitational effects. $\endgroup$ – Hans-Peter Stricker Nov 13 '14 at 20:01
  • $\begingroup$ Why does a tripod have only 3 legs? $\endgroup$ – Prahar Nov 13 '14 at 20:03
  • $\begingroup$ @Prahar: haha! See my comment above. (On an irregular surface with low gravitational forces the tripod example doesn't help further.) $\endgroup$ – Hans-Peter Stricker Nov 13 '14 at 20:05
  • $\begingroup$ Now that the question has been clarified (see Hans' comment to my answer), he is asking about adding a fourth leg pointing upward so that the vehicle would be stable regardless of orientation. $\endgroup$ – David Hammen Nov 13 '14 at 20:06
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Have you ever sat on a wobbly, four-legged chair? That can't happen with a three-legged chair. A three-legged chair is stable. Four legs is one leg too many. A four-legged lander would need to adjust the lengths of the legs to stabilize it.

Update
Aside: The English word you are seeking is tetrahedron. Now that the intent of the question is clear, the answer is no. Adding a fourth leg to Philae so it could land in any orientation wouldn't work because Philae has a preferred orientation. There would be no point in adding the capability to land in some other orientation because many of the experiments on the wouldn't be able to function. Philae has a definite "up" and "down".

Second update
A tetrahedral object can land in four different orientations. Only one of those orientations is useful because Philae has a definite "up" and "down". There would be no point in adding a fourth tetrahedral leg without the ability to reorient the central body of the vehicle. That adds considerable complexity to the vehicle.

The reason Philae had problems was because its simple harpoon system and its simple thruster system failed. BTW, those harpoons and that thruster were part of what gave Philae a definite "up" and "down". As was seen with Philae, even simple stuff has an unfortunately high propensity to fail in spacecraft. Complex stuff? That fails with even greater propensity. Spacecraft designers have learned this lesson many times over and regularly have to tell themselves to "keep it simple, stupid!" The KISS principle is driving concept in designing space vehicles.

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    $\begingroup$ The question was not about four legs as a table has, but about tetrahedron-shaped four-legged object, like methane molecule (with hydrogen atoms as legs). $\endgroup$ – Ruslan Nov 13 '14 at 19:55
  • $\begingroup$ We are not talking about something like a plane underground with strong gravitational effects - but about an unpredictabily irregular underground with some soft gravitational effects. $\endgroup$ – Hans-Peter Stricker Nov 13 '14 at 20:00
  • $\begingroup$ @David: But now one of the three legs rises helplessly into the air. Wouldn't it be better that a fourth leg touches ground? In any case? (Left alone of what you explained - which I hope to have understood.) $\endgroup$ – Hans-Peter Stricker Nov 13 '14 at 20:10
  • $\begingroup$ @HansStricker -- I haven't the foggiest idea what you mean by that, or why you would think that extra legs would help solve that problem. Please clarify the question. $\endgroup$ – David Hammen Nov 13 '14 at 20:14
  • $\begingroup$ Does this mean: the best one can do is to give a lander three legs? (Because we live in three dimensions.) I claim: this is not sooo obvious. $\endgroup$ – Hans-Peter Stricker Nov 13 '14 at 20:25

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