Sol 550 : A MAHLI Panorama

Here is a hotlinked sol 549 Navcam image of a small ridge that captured the interest of Curiosity. I think it’s about 2 feet long:

On sol 550 it took a series of MAHLI closeups, and also this Navcam image of the robotic arm ( with the MAHLI on it ) during the process of doing so:

BTW, here is an image which shows the arm in the stowed position, prelaunch:

There are three joints on the arm itself, with all the axes parallel, so it moves in one plane with motion of these three joints. There’s a vertical “post” that carries the arm assembly and allows it to swing out and away from the base. You can see it projecting downward near the left front wheel.

You can also see the name plate on top of the upper beam of the stowed arm, and the forward projecting arm joint at the right side of the rover. These are visible in many of the downward looking Navcam and Mastcam images, including the first image of this post.

If I had understood all this correctly, I never would have made that blooper in my SPOT OF BOTHER post!

Anyway, the MAHLI took a series of images of this little ridge on sol 550, and they appeared to be suitable for a panorama, so I tried it. It worked pretty well, but it is not a true panorama. The images on the left were part of a “scan” where the MAHLI changed position but kept pointing in the same direction, more or less. Then the images at the right were made by swinging the POV outward. It stitches down the centerline pretty well, but you can see some doubling up of particular features above and below. Well, a nice result and it shows incredible detail. As always, CLICK TO ENLARGE!

Mandelbrot II : Julia Islands

I mentioned the image gallery section of the Wiki Mandelbrot Set article as the starting point for my renewed interest in this topic. It is based on a zoom sequence terminating on a “Julia Island”, which is a component of the Mandelbrot set resembling a “Julia set”, which is based on the actual sequence of an iteration rather than the in/out criterion of the Mandelbrot set. So there’s a little math magic in this resemblance.

Here is my recreation of the zoom, based on the linked tool I mentioned, which by no means outstrips the images in the article ( which can be expanded, ) but it does go a little deeper, and it keeps the Julia Island location at center. This was easy to do by working backwards from the deepest image and keeping the same coordinates by simply changing the scale in the URL field.

Here’s a “hot link” of the lowest level of the Wiki zoom :Wikipedia commons( To see its full glory visit the site. ) It roughly corresponds to the antepenultimate frame of the animated gif. The commentary in the main article notes that there is a “satellite” ( i.e. “mini-Mandelbrot set” ) at the center of the island structure that is “too small to see at this magnification”. Just visible there is the large square structure in the center of the last frame of the gif, where you can almost see the satellite at the center of it, as a tiny dark speck.

This conclusion is supported, if not confirmed, by perusal of a nearby and entirely analogous structure, which is a little bit larger. Here’s a slower zoom, taking 1/2 per frame instead of 1/10, showing a descent through the analogous square structure to reveal the Mandelbrot satellite.

Note the emergence of a new “sea” of similar spirals and structures as part of the INFINITE descent. It’s fascinating and incredible and, like the song says, “a little bit frightening.”

Looking back at Dingo Gap

Here is a 3 X 4 panorama from Mastcam images acquired on sol 538 around 2014-02-09 22:37:19 UTC. (Click to enlarge ! )

Compare the near horizon with this view from the landing site on sol 3:

You can see the same mesa-like structures on the lower reaches of Mount Sharp, and the same low dome to the left of them. Note that the gap between the mesas has opened up from the new position, as Curiosity has moved back and to the right from this sol 3 POV.

Finally, this animated gif is from the Rear Hazcam on sol 538, with images acquired at 2014-02-09 22:03:32 UTC, 2014-02-09 22:56:48 UTC, and 2014-02-10 00:37:23 UTC, so the second of these seems to be the POV of the Mastcam panorama. Note the dome of Mount Sharp visible in the first frame. … A very nice shot!

The Mandelbrot Universe

I recently saw one of several “Deep Zoom” videos of the Mandelbrot set, which I’ll leave alone for now, but it got me interested in the subject, which I haven’t thought about since the 1980’s. Times change, and the facilities available for investigating the subject are now, of course, much more powerful. I looked at the Wikipedia article on the Mandelbrot set, which refreshed and greatly expanded my casual knowledge of the subject. The exciting thing though is the link under the section headed “Image gallery of a zoom sequence” to an “interactive viewer” ( go to Wikipedia to get the link. ) It indicates it will open for “this location” ( i.e. the one illustrated ) but I found it goes to the initial full view of the set.

Anyway, it’s a fabulous tool. You can click on a spot to zoom by a factor of two, and although it does take time to calculate the image, you can click again for repeated zooms without waiting for the full resolution. I found it very handy to use. It goes down to 10^-14 in scale, which is well short of the claims made by these deep zoom videos, but it provides ample scope for casual exploration.

Note the parameters are passed in the URL address field, and these can be edited and entered with the naturally hoped for result. Of course you can save them and reenter them to bring back the image, so as I say, very handy.

I also used the extended precision calculator, bc, on Cygwin to explore some of the mathematical aspects, which I’ll get to. First, here’s an animated gif of a short zoom on a large portion of the Mandelbrot set, progressing down the scale of the circular “bulbs” along the real axis. This is a more extensive version of the “Mandelbrot zoom” gif in the Wiki article which uses a loop of two circles without “hair”.

First note that the Mandelbrot set itself is the black area, determined by points c, where the iteration of the defining formula always remains bounded. The “hair” is exterior to the set and is colored according to how rapidly the formula diverges under iteration.

These circles are in the set, and are called “period bulbs” in the article, because of the defining nature of the points within them. That is, they are points c, in the imaginary plane, such that iteration of x = x^2+c ( where ‘=’ means assignment, as in a programming language ) starting with x == 0 ( ‘==’ asserts equality ) converges to a cycle of fixed values, these values being 2 in number in the “2” bulb ( the first circle ) then 4 in number, then 8, 16, 32, and 64 in number in the large circle of the sixth and last frame.

This brings us to the math, and bc. Let’s go for the gusto with the “64 bulb”. Using the “click and zoom” facility in the interactive viewer, we go to the sixth bulb, i.e. the 64 bulb, and simply copy and paste the x-axis ( i.e. real part ) value into bc, and initialize x to 0:

c= -1.400968738408754

Then we iterate for a goodly number of times to assure ourselves of convergence to a stable cycle, in the belief this must happen:
( Even though bc runs interpretively, the power of a modern PC is such that this number of iterations causes a barely perceptible hitch in the response )

for( i=0 ; i<n ; i++ ) x=x^2+c;x

Now we check for a 64 cycle of values:

for( i=0 ; i<n ; i++ ) x=x^2+c;x

OK! But maybe this is a repetition of a shorter cycle, so check 32:

for( i=0 ; i<n ; i++ ) x=x^2+c;x
for( i=0 ; i<n ; i++ ) x=x^2+c;x

OK again!  We can now be satisfied that 64 is the true cycle value. Note that we didn’t use a lot of precision, but the “attractive cycle” is strong medicine, and we don’t need it. We are just checking the fundamentals though, and of course there are more delicate arrangements to be found.

In particular, what about the “vanishing point” of this self-similarity? It is cited in the article to crude precision as -1.401155, which can be discerned numerically, but what are its defining properties? This point remains obscure to me.

Misiurewicz points

These are defined as values of c such that k iterations of x=x^2+c give a value that repeats after n ( more ) iterations, and are designated as M_k,n, and these form vanishing points for self-similar transformations although they do not account for the seemingly fundamental one just described. Noteworthy is M_4,1  described in the Wiki article.

Here is a zoom which goes by steps of 100X magnification ( after an initial step of 10X )
The Wikipedia article on Misiurewicz points mentions … “(Even the `principal Misiurewicz point in the 1/3-limb’, at the end of the parameter rays at angles 9/56, 11/56, and 15/56, turns out to be asymptotically a spiral, with infinitely many turns, even though this is hard to see without maginification.)” ( sic ! just noticed that. ) … This is our guy, and you can see the “slow roll”, which advances by about 50 degrees in 6 steps of 100X magnification. ( I’m still working on the meaning of those angle values. )

In this case we can do a little math and come up with a way to define the M_4,1 point, at the center of the zoom, to arbitrary precision. Approaching it graphically we see the center at approximately ( -.10109636 , .95628651 ) in the complex plane.

In bc we define functions:

define f(a,b){ return( a^2-b^2 + c); }
define g(a,b){ return( 2*a*b +d ) ; }

… which are the real and imaginary parts of the formula, x^2 + (c,d) , applied to a complex value x with real and imaginary parts (a,b) . So we can proceed to apply the formula iteratively and see if we come up with a constant value after 4 iterations, as per the M_4,1 designation:

c= -.10109636
d= .95628651
z=f(x,y);y=g(x,y);x=z;x;y # iteration 1
z=f(x,y);y=g(x,y);x=z;x;y # iteration 2
z=f(x,y);y=g(x,y);x=z;x;y # iteration 3
z=f(x,y);y=g(x,y);x=z;x;y # iteration 4
z=f(x,y);y=g(x,y);x=z;x;y # iteration 5

Note that the 4th iteration approximately achieves the repeated value, but the 3rd iteration is the negative of it. So to home in on the exact value we want to adjust c so that f(f(f(f(0)))) = – f(f(f(0))). Without spelling it all out, we can do this by numerically evaluating the sum of the 3rd and 4th iterate ( which we want to be 0 ) to see how it varies with small variation in the real and imaginary parts of c . This is essentialy “Newton’s method” of finding roots. Inverting a simple 2X2 matrix tells how to vary ( c, d ) to bring this sum to 0. By iterating this estimation we can find  the correct complex value of M_4,1 to arbitrary precision.

Here’s as far as I’ve taken it so far:
( c, d ) = (
#real part
# imaginary part

It’s very easy to do and could be taken to many thousands of decimal places in less than an hour, I’m sure.

I’ve got a lot more, but I want to put something out now and I do plan to continue my discussion … but let me just outline my thoughts.

The mathematical structure of the Mandelbrot set has been a matter of ongoing investigation, and is a difficult subject. One fundamental point is the “connectedness” of the set, and it has been proven that it is “connected” according to the literature. Well, I’m sure it has, but I still have questions! In particular it seems to me that these Misiurewicz points must be isolated elements of the set. Perhaps in some sense they are, since topology has more definitions than you can shake a stick at.

Anyway, I hope to be back with more pictures.

A spot of bother

As publicized, it appears Curiosity has made its move to traverse the rather striking looking sand bar at Dingo Gap. However, it looks like it is hung up a little. The panorama below, made from sol 533 Mastcam images, shows that the stowed robotic arm, with the nameplate on it, has touched the sand on the far side of the dune, which is just a few feet across here.

BTW, to make this panorama “stitch” I had to manually add control points by matching the tiny pebbles in the sand, so I’m quite proud of myself!

UPDATE: This “hangup” is just an illusion caused by the featureless sand background. The appearance I mistook for marks in the sand is actually strapping holding the nameplate to the round support arm. The support arm itself has the appearance of a shadow on the sand leading to my erroneous “gestalt” perception.

Here is a Navcam view from the same position which has a similar appearance, but in this case we can see that the drill bit holder ( attached to the front of Curiosity ) is beneath the nameplate, so the nameplate and arm are well clear of the sand.

Here is an animated gif from the sol 533 Front Hazcam images showing the approach to the crest of the dune.

The Mastcam panorama was taken around 2014-02-04 18:37:37 UTC whereas the last frame of the gif was at 2014-02-04 18:07:27 UTC, and the sol 534 Front Hazcam view is unchanged from sol 533, so I’m not sure how that works out.

… well, this just in: According to the sol 535 Front Hazcam view, shown here as an anaglyph, it looks like Curiosity has cleared Dingo Gap!

I’m not sure they’ve reached the promised land of no rocks, though.