SpaceCraft User Manual
Luby Prytulak PhD
Version of 30 Aug 2015 12:10pm PDT
Most current versioin of the SpaceCraft computer program is spacecraft-012.exe
You are at: http://www.twelvebytwelve.net/spacecraft/spacecraft-user-manual.html
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The SpaceCraft User Manual is divided into two parts — Part I: THE THEORY BEHIND SPACECRAFT and Part II: ACTUALLY PLAYING SPACECRAFT. Part I will be of greatest interest to researchers, academics, and educators. Anyone more interested in actually beginning to play is able to go directly to Part II by clicking here.
If you send me a video of the six faces of the SpaceCraft cube being solved in under 30 seconds, I would be delighted to post it online, under a PlayerName of your choice.
Part I : THE THEORY BEHIND SPACECRAFT
IT BEGINS WITH JEAN PIAGET
Jean Piaget (1896-1980)
In their book The Child's Conception of Space, Swiss developmental psychologist Jean Piaget and coauthor Bärbel Inhelder describe their study of one hundred children, ranging in age from 4 to 12, who were asked questions concerning the Three-Mountains pasteboard model sketched below.
The smallest mountain was green, and topped by a house. Next-highest was a brown mountain topped by a red cross. The tallest mountain was gray, and topped by a snowcap. The model sat on a one-meter square platform, with mountain height ranging from 12 to 30 centimeters.
Each child would view the model from one of the positions A, B, C, or D. In one of the tasks, a wooden figurine, 2 or 3 cm high, sometimes referred to as the "little man" or the "doll" (and here referred to most often as "LittleMan"), would be placed in a position different from the child's, and the child asked to pick which from a collection of pictures of the scene is the one that LittleMan sees (sometimes expressed as which would be the photo that LittleMan would take if he had a camera). In an alternative task, the child starts with a picture of the scene taken from a perspective other than his own, and is asked where LittleMan should be placed so that he will see what is in the picture.
Piaget, J. and Inhelder, B. The Child's Conception of Space, Routledge & Kegan Paul, London, 1967, p. 211.
Added to the original Figure 21 has been color, and the words "View from A".
The younger children proved to be utterly incapable of arriving at the correct answers, most typically attributing to LittleMan the same perception as their own, a deficit which Piaget attributes to perceptual egocentrism or to the egocentric illusion, which he describes as follows:
[I]t is found that the child fails to realize that different observers will enjoy different perspectives and seems to regard his own point of view as the only one possible. p. 213
Instead of constructing the perspective corresponding to the different positions, the child considers his present point of view the only possible one and is unable to deduce from it the transformations produced by a change of position. p. 215
The children [...] all really imagine that the doll's perspective is the same as their own, they all think the little man sees the mountains in the way they appear from where they themselves sit. p. 220
[I]f confronted with [...] the group of mountains [...], he appears to be rooted to his own viewpoint in the narrowest and most restricted fashion so that he cannot imagine any perspective but his own. Indeed, he cannot imagine any perspective but that of the passing moment, since with a change of [his own] position he repeats his performance in terms of the new position! pp. 242-243
Piaget, J. and Inhelder, B. The Child's Conception of Space, Routledge & Kegan Paul, London, 1967.
Although with advancing age the children began to abandon their perceptual egocentrism, they did not achieve complete mastery of the Three-Mountain tasks until about the ages 9 or 10.
Success in Piaget's Three-Mountains tasks, then, depends upon the skill of visualizing three-dimensional scenes from different perspectives. It proves to be the case that an important component of this skill is mastering the concept of Left-Right.
THE IMPORTANCE OF LEFT-RIGHT IN THE THREE-MOUNTAINS TASKS
Suppose a child is placed at A in the display below, and LittleMan is placed at C. This particular LittleMan happens to have his right hand raised, and a question mark emblazoned on his chest as if asking "What do you think I see?".
|View from A
Let us now imagine the child being asked to select which from a collection of photos represents what LittleMan sees. The child who realizes that LittleMan sees the lightbulb and mug lying right in front of him, and the house lying behind, is still faced with the difficulty that this is equally true of two of the photos found among those from which he is asked to choose:
Lightbulb and mug are close to LittleMan, and the house lies behind,
so is this LittleMan's view from C ?
Lightbulb and mug are close to LittleMan, and the house lies behind,
so is this LittleMan's view from C ?
To get the correct answer, the child can start by identifying LittleMan's right hand, which makes possible the observation that the lightbulb is on LittleMan's right, and which justifies the conclusion that it is the first of the two photos above that represents what LittleMan sees, because that's the photo showing the lightbulb on the right.
Inferring what LittleMan sees, then, starts with identifying his right hand, and where his having it raised is an accident of the present case, and cannot be depended on in the future. And neither is the head of the question mark drooping to LittleMan's right to be relied upon, as any future LittleMan is likely to be lacking a question mark, and lacking any other extraneous cue signalling left-right.
Understanding how relationships reverse, as for example between View A and View C, then, is the skill needed to infer what LittleMan sees, there being two such reversing relationships:
The in-front-of vs. behind relationship is easier to understand, and is mastered earlier; the left vs. right relationship is more difficult, and is mastered later; this not without reason:
- the in-front-of vs. behind relationship which reverses from View A (where the child sees the bulb and mug as farther) to View C (where LittleMan sees the bulb and mug as nearer), and
- the left vs. right relationship which also reverses from View A (where the child sees the bulb on his left) to View C (where LittleMan sees the bulb on his right).
Along with these correct solutions, however, past errors are repeated [...]. In most cases they occur in connection with relations of left and right, before and behind being reversed with less difficulty. [...] The reason for such a tendency is fairly clear inasmuch as [...] there is a bigger difference between a background beyond the reach of immediate action and a foreground directly subject to it [which is within the reach of immediate action], than there is between a left and right which are equally near or distant. Hence the before-behind relationships become [...] more responsive to changes of perspective, sooner than those of left and right.
Piaget, J. and Inhelder, B. The Child's Conception of Space, Routledge & Kegan Paul, London, 1967, pp. 235-236.
THE IMPORTANCE OF LEFT-RIGHT IN OTHER SPATIAL TASKS
If being able to solve some spatial tasks requires being able to identify left-right, then the solution may be assisted by imagining a LittleMan.
Take, for example, the four objects on the left below, which we will refer to as "Squiggles". One question that comes to mind is whether all four Squiggles have identical shapes (we don't care about colors). A second question is whether pairs of Squiggles can me mated so as to form a 2x2x2 cube. And if some pairs can form cubes, and other pairs can't, then which ones?
One might say these Squiggles have extremely simple shapes, and the questions asked about them are extremely simple as well, such that any adult should be able to answer them after a moment's reflection, and yet I expect that almost no adults will be able to answer them. I certainly wasn't able to until I gave the matter a great deal of thought — the great deal of thought that tried to make up for my total lack of training in spatial skills, or one might say total lack of training in space craft, written here as SpaceCraft, in which "craft" intends to convey not so much "vehicle" as "skill", as it does in stagecraft, statecraft, or witchcraft.
What everyone does see immediately is the strong resemblance of one Squiggle to another — each Squiggle is composed of two beams, each beam measuring 1x1x2 units, and so each squiggle has a total volume of 4 cubic units. And the Squiggles are also identical in that each of its beams attaches to the other in what looks like the same way. But how to more accurately describe "the same way"? One way to describe the attachment is to notice that it produces, on two sides of each Squiggle, an L shape, each L having an area of 3 square units. It might be possible for the reader to locate the two sides of each Squiggle that do take on the shape of an L.
Here they are, all the Ls. The Brown Squiggle is sitting on one of its Ls, as indicated by "L base"; its other L is hidden — it stands vertical and facing the NorthEast. The Yellow Squiggle on the left is also sitting on one of its Ls; its other L is hidden — it stands vertical and facing West. Orange is sitting not on an L, but rather on one of its 1x2 sides; both of its Ls stand vertical; one is hidden and facing NorthEast, and the other L is visible — and facing SouthEast. The remaining Yellow Squiggle is atilt; one of its Ls can be seen pointing NorthWest, its hidden L faces NorthEast.
Although the question of whether all Squiggles are identical remains unanswered, the second question is easier to answer, so let's answer it first: Can pairs of Squiggles be mated to form a 2x2x2 cube? Trying to mate the two Yellows fails, as does trying to mate the Orange with the Brown. Maybe it will prove to be impossible to create a cube?
And mating one of the Yellows to Orange, and the other to Brown, fails as well. It's beginning to look like creating a cube is impossible?
But wait! If Brown and Red swap partners, we do get cube creation. The two Yellows, it seems, do not have identical shapes. Squiggles generally, then, are not identical. And nobody said that Squiggles of the same color had to have identical shapes, or that Squiggles of different colors had to have different shapes. What we have discovered, as will be proven below, is that Squiggles have to have identical shapes in order to cube.
To demonstrate exactly how Squiggle shape differs, we position the four of them following two rules: RULE 1. One of the Ls must lie flat on the ground, allowing us to view each Squiggle as a building consisting of a 1-unit-high low segment having floor area 1x2, and with a 2-high tower, floor area 1x1, attached. Since it helps to think of the Squiggles as buildings, let us begin calling them Buildings.
RULE 2. We always position ourselves so as to view every Building with its low part closer, and with its tower behind. Let us call any view which meets these two conditions a StandardView. Having achieved a StandardView for all four Buildings, then, what we see is that Buildings A and B can be called "TowerLeft", and that Buildings C and D can be called "TowerRight". Our mating rule now becomes: two TowerLeft Buildings mate into a cube, and two TowerRight Buildings mate into a cube, but a TowerLeft and a TowerRight Building are incapable of mating into a cube.
If we begin to rotate our collection of Buildings, and a LittleMan (with raised right arm) follows along, repositioning himself so as to maintain a StandardView, his perception will continue to reassure him that Buildings A and B are TowerLeft, and Buildings C and D are TowerRight.
If we spin our Buildings 90° CounterClockWise (CCW), LittleMan is able to reposition himself so as to maintain a StandardView, and as a result does not waver from his conviction that Buildings A and B are TowerLeft, and Buildings C and D are TowerRight.
Anogther 90° CCW spin, and LittleMan continues to reposition himself into the StandardView, and so identifies the Buildings as before.
And a final 90° CCW spin of the Buildings, and another repositioning of LittleMan into the StandardView, and another confident identification of the Buildings in conformity with their previous identification.
And now comes a demonstration of how it is possible for us to be presented with a Squiggle in any orientation whatever, as for example that above, and to quickly identify it as either TowerLeft or TowerRight. We start by thinking of the Squiggle as a Building, and locating its Ls.
Here are the two Ls, one happending to be facing SouthEast, and the other SouthWest.
Starting with L SouthEast, we imagine a pane of glass that coincides with it extending outward in all directions, and with LittleMan standing on it, as if it was the ground on which the Building stands, which satisfies RULE 1. And LittleMan sees the low part of the Building nearer him, and the tower farther away, which satisfies RULE 2. And as the tower is on LittleMan's right, the Building can be identified as TowerRight.
Or, with the Building still in the same orientation to us as before, we can imagine LittleMan placing himself in the alternative StandardView, the one in which L SouthWest defines the ground on which LittleMan stands, which satisfies RULE 1. And LittleMan sees the low part of the building in front, which satisfies RULE 2. From this alternative StandardView, the tower is again on the right, so the identification of the Building as TowerRight is confirmed. As relying on one L leads to the same result as relying on the other, relying on only one of them is sufficient to correctly identifying Squiggle shape.
The above-described procedure should make it possible to go back to the first photo of our four Squiggles, and fairly quickly identify each of them as either TowerLeft or TowerRight. The procedure should also make it possible to go back to the three photos showing either success or failure to mate into a cube, and to identify each of the Squiggles in the attempted matings as either TowerLeft or TowerRight, and while doing this to verify the rule that cubing is possible only when Squiggles have the same shape.
This does not mean that in order to discover whether some particular Squiggle is TowerRight or TowerLeft, it is necessary for us to carry a plastic LittleMan around with us. LittleMan represents where we will be imagining ourselves to stand. It may be expected that with practice, what at first seems like a laborious mental process becomes easy, and eventually falls away altogether, leaving the practicer with the ability to identify the left-right of LittleMan, no matter how strangely oriented, and with the same facility as identifying LittleMan's up-down, or front-back; and may be expected to leave the practicer also with the same facility of identifying TowerLeft and TowerRight in Squiggles; but most importantly, may leave him with a generalized skill of being able to mentally visualize all manner of complex objects from the whole gamut of perspectives from which the objects can be viewed.
FROM CHRONICLING TO TEACHING
Within Developmental Psychology, Jean Piaget plays the valuable role of chronicler of spontaneous learning.
By "spontaneous learning" is meant learning that takes place without formal instruction. That is, the spatial skills that Piaget examines seem not to be part of his children's school curriculum, and seem absent from schooling everywhere. Undoubtedly, the development of spatial skills depends upon experiences with space, but what experiences contribute to what learning can only be guessed.
And to say that Piaget plays the role of chronicler of such spontaneous learning is to notice that he describes and he theorizes, but he does not teach. He neither tests the efficacy of classroom exercises, or learning experiences of any kind, which might improve performance, nor even contemplates or recommends any. It would be out of character for Piaget to voice dissatisfaction with the slow pace of a child's spontaneous acquisition of skills of any kind, and out of character for him to recommend speeding that acquisition, or deepening it.
Although the SpaceCraft program follows in Piaget's footsteps, one way in which it departs from Piaget precedent is that it provides a tool for advancing the acquisition of spatial skills, and deepening the skills acquired. SpaceCraft supplements spontaneous learning with deliberate training, with the expectation that what will happen to spatial skills is what happens when deliberate training supplements spontaneous learning in any field, say in mathematics.
In mathematics, a twelve-year-old who has never gone to school or received any other form of instruction might pick up not much more than addition and subtraction of small integers, whereas sophisticated formal instruction is able to teach him Algebra and Trigonometry and Calculus, as can be seen in the case of Marko who at the age of 12 years and 7 months received a 95% final grade in a course in which the final examination looked like this:
The question before us is whether spatial skills resemble mathematical skills — that is, whether the spatial skills bestowed by spontaneous learning are the equivalent in mathematics of adding and subtracting small integers, and the spatial skills bestowed by deliberate instruction might prove to be the equivalent of doing Algebra and Trigonometry and Calculus?
But what would training in higher-level spatial skills be like? It could begin with the recognition that the Piagetian Three-Mountains tasks are a small selection from a much larger number of tasks that are available. Consider, for example, that with the child at A, LittleMan could be placed at C not standing, but lying on his left side (and to get him off the ground, let us place a small bunk bed at C and have LittleMan lying on the upper bed, his left ear pressed to the pillow). Now, what would LittleMan see? It would be this:
Or, LittleMan could be lying on his right side on that upper bunk, or he could be doing a handstand on it.
And, at the same time, assuming that the bulb-mug-house are glued to a platform, the platform could be rotated every whichway as well — say tilted so as to stand on its edge, or suspended upside down from the ceiling. If the platform were made of glass, the three objects could be pointed away from the child, so as to be visible only through the glass.
The number of possibilities is staggering, and the complexity may seem pointless, but then so does Algebra and Trigonometry and Calculus seem both staggering and pointless to one whose math is limited to the kindergarten arithmetic picked up by means of spontaneous learning.
The immediate goal, then, might be to begin building spatial skills which are not tied to a narrow range of conventional environments. One might say that the ultimate goal of SpaceCraft training is to prepare the student to function efficiently in weightless space. The ultimate goal, then, may be thought of as astronaut training. SpaceCraft equips students with skills necessary for survival in outer space.
But, it may be asked, given that almost nobody is going to be an astronaut, why bother? We seem to pick up all the spatial skills we need spontaneously. What's the point of rushing a child toward a goal that nature supplies at its own gentle pace?
Among the answers to this question are the following six:
(1) The younger the age at which something is learned, the deeper and more permanent is that learning, as everyone who enjoys some familiarity with non-Anglo immigrant families sees evidenced — the youngest members of the family learn to speak grammatical and unaccented English, while the oldest make little progress. And in between the two extremes, it is around age fourteen that language acquisition can be seen to have markedly slowed. The announcement that the learning of English began after age fourteen blares from every sentence uttered for the rest of the late-starter's life.
(2) Children are hard-wired to learn voraciously and gleefully. They hate school because it suppresses voracious learning, and inculcates passivity instead. To provide children with the means to learn voraciously is to satisfy their hunger for learning. Children can learn advanced mathematics, or advanced spatial skills, with the same ease and lack of stress as they become tri-lingual, as for example acquiring fluency in French at home, in Italian while summering with mother's family in Florence, and in English at school.
(3) Advanced spatial skills are needed to understand science, and to contribute to scientific progress, as will be glimpsed in examples from Physics, Chemistry, and Biology further below. Science is so complex, and so vast, that one has to start early to make inroads.
(4) That some occupations require advanced spatial skills may not be widely recognized, as will be glimpsed in reference to air traffic control further below.
(5) Spontaneous learning produces many adults who lack even such elementary spatial skills as Piaget tests, as will be testified to further below.
(6) Advanced spatial skills are being acquired even with no assist beyond spontaneous learning, and even without the awareness of the individual. To promote the learning of advanced spacial skills is no more than to facilitate what already happens spontaneously to some degree, as will be illustrated immediately below.
Let us take a quick look at each of the last four above points in reverse order, starting with the unconscious acquisition of advanced spatial skills.
Elaboration of Answer (6): SPONTANEOUS LEARNING CREATES UNSUSPECTED SPATIAL SKILLS
When I recently invited four people at a gathering to try reading text that was upside-down, I found that two of them were able to do it, though somewhat haltingly, which was already a great surprise. The greater surprise was that the two others were able to read upside-down at high speed, just as if they were fluent readers reading text that was upright. And the greatest surprise was that one of the very-fast upside-down readers had all his life disliked reading, and had gone out of his way to avoid it as much as possible.
It seems also that if one can read text rotated 180° from upright (which is upside-down), then one is also able to read text rotated 90° or 270°. Put more broadly, it is possible that a large proportion of the literate population is able to read text presented at any orientation whatever, such that SpaceCraft presenting scenes whose components may lie in any orientation whatever is only presenting scenes that man can easily process using skills implanted in him by their having favored survival throughout his evolution.
Elaboration of Answer (5): SPONTANEOUS LEARNING ALSO LEAVES SPATIAL-SKILL DEFICITS
At the same time that spontaneous learning leaves many with advanced spatial skills, it leaves others with deficits:
Which Is Right?
by Belle Elving, Washington Post, 22 Jul 2008
I can't tell right from left.
It hasn't been a serious problem. Except that night on a freeway heading into San Francisco when, befuddled by an "Exit Left" sign, I hit the brakes and got totaled by a really fast sports car. Or the day I directed a footsore family of tourists 180° away from the White House. Or the time I assembled an Ikea bookcase with the dowel holes for the shelves on the outside. Or the countless times I've annoyed my husband by telling him "Turn, um, left. No wait, I'm sorry...."
It's a mild disability that has not seriously limited my options in life. Of course, a career in air traffic control would have been unwise. Synchronized swimming and ballroom dancing were not in the cards. (Playing cards is a bit of a problem, actually. I'm never sure which way to pass them.) But I'm usually fine driving alone. I know which way I want to turn; I just don't know what to call it.
On the upside, it's delightful to discover others who share this condition, including, as it happens, the editor of the Health section and the editor who wrote the accompanying medical misadventure story. And we are not that small a group. John R. Clarke, a professor of surgery at Drexel University in Philadelphia, estimates that about 15 percent of the population faces some degree of left/right challenge. Eric Chudler, a neuroscientist at the University of Washington in Seattle, puts the figure a bit higher, having found that more than 26 percent of college students and 19 percent of college professors acknowledge having difficulty telling left from right — occasionally, frequently or always. [...]
Thus, SpaceCraft is available not only as a diagnostic tool able to detect and measure deficiency, but also as a learning tool which may be able to correct it.
Elaboration of Answer (4): OVERLOOKED SPATIAL-SKILL REQUIREMENTS
Belle Elving's reference above to air traffic control as requiring advanced spatial skills may serve as a reminder that such may be the case in many professions, but which may not be widely known simply because people generally are not closely acquainted with just what skills are called for in most professions. In the case of air traffic control, the diagram below gives some idea — among the skills required is being able to accurately and unerringly visualize an airplane's location within a complex three-dimensional space, something the pilot needs to be able to do along with the air traffic controller:
World Book Encyclopedia, World Book Inc., Chicago, 1985, Vol. 1, p. 236.
Elaboration of Answer (3): SPATIAL SKILLS IN PHYSICS
The first topic treated in a first-year-university Physics course is likely to be Kinematics, which is the the study of objects in motion, and among the Kinematics problems is bound to be one of a boat plowing through a moving river, like the following. (The solution is not germane for our purposes, and is not shown.)
John D. Cutnell and Kenneth W. Johnson, Physics, John Wiley & Sons, New York, 1989, p. 61.
At first glance, it may seem that this problem has nothing to do with the tasks Piaget presented children in his Three Mountains study, and yet it can be shown that this Kinematics problem is a more complex version of the Piagetian tasks, one that has introduced velocity and quantification, but which nevertheless subsumes within itself the question of what LittleMan sees.
The situation is that a boat is aiming straight across a river (downward in the illustration) and travelling at 4.0 meters per second (m/s), while the river is flowing downstream (rightward in the illustration) at the rate of 2.0 m/s. The vertical red arrow shows what would be the velocity and path of the boat if the river stopped flowing and the boat moved on its own power. The horizontal red arrow shows what would be the path and velocity of the boat if its motor were turned off and it just drifted in the flowing river. The black arrow shows what will be the path and velocity of the boat with its motor running and the river flowing.
Were this task reduced to Piagetian proportions, three questions could be asked:
(1) If LittleMan was floating in a rubber raft at the tip of the vertical red arrow, what would he see if he stared at the prow of the boat?
Answer: LittleMan would see the prow coming straight at him at a velocity of 4.0 m/s.
(2) If LittleMan was a passenger in a car driving at 4.0 m/s across the river on a bridge following the vertical dashed line and alongside the boat, and he spent his time staring out of the side window of the car, straight upstream, what would he see?
Answer: LittleMan would see the boat from the side, but nevertheless approaching straight toward him at a velocity of 2.0 m/s.
(3) If LittleMan was sitting on a rock poking out of the water, situated at the bottom of the large black arrow, and if he stared directly at the prow of the boat, what would he see?
Answer: LittleMan would see the prow approaching him at a velocity of 4.5 m/s. Despite the prow appearing to always point left of LittleMan's left shoulder, its approach would be linear so that if he was doing his staring through binoculars initially aimed directly at the prow so that the prow was exactly in the center of the field of view, let us say directly on the crosshairs of the binoculars, assuming it has crosshairs, and if the binoculars were locked into that position so that they were unable to swivel during the boat's voyage, LittleMan would nevertheless find that the prow he was staring at approached him without wandering off the crosshairs. Binoculars similarly locked in the first two situations would also show the prow not wandering from the crosshairs, indicating that the approach of the prow was linear in all three cases.
The three vectors of the boat problem, then, can be understood to simultaneously reflect the perceptual experiences of a person viewing the situation from three different points of view. Kinematics problems do not often ask the question of "What would LittleMan see if...?" because the quantitative questions that they do ask assume that the corresponding LittleMan questions could easily be answered. If this is not a reasonable assumption — if in fact some students learn to answer the quantitative problems without being aware of their LittleMan implications — then perhaps their understanding is more superficial than it should be, and needs to be enriched by learning to answer the LittleMan questions along with the quantitative ones.
And the awareness of left- or right-handedness that is so integral to Three-Mountain tasks can be seen to be needed throughout physics, as is exemplified in the three right-hand rules illustrated below.
||Figure 9.12 illustrates the Right-Hand Rule for Angular Acceleration. (p. 185)
Figure 27.7 illustrates the Right-Hand Rule for Magnetic Force. (p. 562)
Figure 27.20 illustrates the Right-Hand Rule for Current-Carrying Wire. (p. 575)
John D. Cutnell and Kenneth W. Johnson, Physics, John Wiley & Sons, New York, 1989.
Further elaboration of Answer (3): SPATIAL SKILLS IN CHEMISTRY
To master concepts related to right- and left-handedness assists in the mastery of the concept of chirality in chemistry.
The Wikipedia page titled Chirality (chemistry) offers the following explanation of chirality, along with an example:
A molecule is considered chiral if there exists another molecule that is of identical composition, but which is arranged in a non-superposable mirror image.
Human hands are the most universally recognized example of chirality: the left hand is a non-superposable mirror image of the right hand; no matter how the two hands are oriented, it is impossible for all the major features of both hands to coincide.
In chemistry, chirality usually refers to molecules. Two mirror images of a chiral molecule are called enantiomers or optical isomers. Pairs of enantiomers are often designated as "right-" and "left-handed".
Two enantiomers (or optical isomers) of a generic amino acid that is chiral.
Still further elaboration of Answer (3): SPATIAL SKILLS IN BIOLOGY
The Wikipedia subject is Nucleic acid double helix. The caption for the illustration below is "The structures of A-, B-, and Z-DNA". To be noticed is that each of the three types of DNA has the geometric attribute of "Helix Sense" which can be either "right-handed" or "left-handed". Not shown, but also to be taken into account, is that the total number of different DNA structures is not limited to A, B, and Z, but rather at the moment numbers 22, leaving only four of the letters A to Z unused. Also not shown but worth keeping in mind is that the DNA helix is not always double, but also can be triple or quadruple.
The moral to be drawn from all of the above is that the structure of DNA is immensely complicated, and that its pieces fit together in three dimensional space, and where the exact angle of fit of every part can be calculated and is known. Among the extraordinary skills that the discoverers of these wonderful structures had to possess are extraordinary spatial skills. And to understand these discoveries requires extraordinary spatial skills. And to venture forth to make new discoveries in this realm will take extraordinary spatial skills.
That the structures are each categorized as right-handed or left-handed is one very small detail in this picture, but a detail important from our point of view because it reminds us that the left- and right-handedness that SpaceCraft teaches has its analogues or counterparts or echoes throughout physics and chemistry and biology, and indeed throughout all of science.
|Helix Sense: right-handed
||Helix Sense: right-handed
||Helix Sense: left-handed
Wikipedia: DNA orbit animated.gif
And it is not just two-dimensional spatial skills that are required, as if DNA could be adequately represented flat on a piece of paper. It is three-dimensional spatial skills that are required. It is the ability to visualize DNA in three-dimensional space, and the ability to rotate that visualization in one's mind, and inspect it from different viewpoints, and not only from different sides, but also, for example, from directly above or below, as was being done in the three circular patterns above.
This is an example of the sort of advanced spatial skill toward which Piaget described an early step — the early step of learning to imagine what the Three Mountains look like to LittleMan stationed at vantage points A, B, C, or D.
Part II : ACTUALLY PLAYING SPACECRAFT
USE SIX ROTATIONS TO PLAY SPACECRAFT
Playing SpaceCraft requires a solid understanding of the six rotations that any object (be it an airplane or a human head or a human body) is able to perform.
The six rotations are Pitch Up and Down, Yaw Left and Right, Roll Left and Right, as illustrated below. It will be noted in the Roll diagram that Left and Right refer to the pilot's Left and Right:
Adapted from D.L. ENGINEERING at
The six rotations take place around three axes — the Pitch Axis, the Yaw Axis, and the Roll Axis. The blue arrows below show the rotations Pitch Down, Yaw Left, Roll Left:
Adapted from an image available on several sites on the Internet, and sometimes attributed to
Robert DeChristopher, as for example at
Suppose that we flipped the above image so as to represent an airplane flying above us, on a horizontal path but upside down, and with the blue arrows unchanged. Of course left and right would continue to be the pilot's left and right, so that the Yaw arrow would continue to point to Yaw Left, and the Roll arrow would continued to point to Roll Left. But what about Pitch? The Pitch arrow indicates that the nose of the plane would point higher above the surface of the earth, which might suggest that the Pitch should now be Pitch Up. However, this is not the case. The reference for Pitch Up and Pitch Down is not gravity and it is not the surface of the earth, but it is the pilot himself. Pitch Down is what the blue arrow points to below, because Pitch Down is when the pilot's nose rotates around the Pitch Axis as if attempting to reach his toes. (Pitch Up would then have been the pilot's nose rotating around the Pitch Axis as if attempting to escape his toes.) The Pitch shown below, then, continues to be Pitch Down, even though the airplane would be aiming its nose farther away from the surface of the earth, and even though the airplane would begin climbing higher above the earth.
Arrows still point to Pitch Down, Yaw Left, Roll Left
Below is a NASA illustration showing the same three axes of rotation, and whose red arrows happen to point to Pitch Up, Yaw Right, and Roll Right:
goes on to explain how the airplane's elevator (consisting of the two black flaps at the rear) controls Pitch Up and Pitch Down:
explains further how the rudder controls Yaw Right and Yaw Left:
shows how the ailerons control Roll Right and Roll Left:
The same six names can be used to describe rotations of the head, the three arrows below identifying three of the six rotations:
Image adapted from a Leeds University medical examination at mcqs.leedsmedics.org.uk/~
The same three axes are used in SpaceCraft, but with their origin shifted downward from the middle of the head (as shown above) to the center of gravity of the entire body (as shown below). That is, where the Roll Axis attached to the lady's nose (see above), now it attaches to approximately ROBO's navel (see below), AND where the Pitch Axis attached to the lady's ears (see above), it now attaches approximately to ROBO's hips (see below).
The Pitch Axis connects to the hips. Mnemonic — Pitch Up means the nose is trying to get away from the toes.
If we imagine ROBO walking down a street, and snagging his toe in a crack in the sidewalk, and falling forward, then we would say this resembles Pitching Down. If he leaned against the hood of a car behind him to look up at an airplane, we would say that he was Pitching Up.
A back flip on the trampoline approximates Pitch Up 360°.
The Yaw Axis connects to the top of the head. Mnemonic — Yaw Right means the nose is trying to point in the same direction as the right shoulder.
If ROBO turned to look at someone standing on his right, we would say that he had Yawed Right; and at someone standing on his left, we would say that he had Yawed Left. What in everyday speech we would call rolling over in bed, say to the right, our technical language would characterize as Yawing Right. A ballerina pirouetting on her toe is Yawing, as is a skater spinning on one skate.
The Roll Axis connects approximately to the navel. Mnemonic — Roll Right means the right ear is trying to lie down on the right shoulder.
As the face of the gymnast below is toward us throughout, we understand that it is his left hand that hits the ground first, and so we say that his cartwheel approximates Rolling Left.
Wikipedia page on Cartwheel (gymnastics)
ROBO is controlled by tapping six keys on the numeric keypad to produce a 45 degree rotation in each of the six directions defined above.
|7 PITCH UP
|8 YAW RIGHT
|9 ROLL RIGHT
|4 PITCH DOWN
|5 YAW LEFT
|6 ROLL LEFT
Blue font indicates the right-hand finger that should be used to strike each key. If occasion ever demands, keyboard keys directly above and below those shown should be struck with the finger assigned to their column. Keys farther right than the ones shown above, like the [ENTER] key (which on some keyboards is designated the [RETURN] key), should be struck using the fifth (baby) finger. Note that in the full keyboard diagram further below,  on the keyboard is shown being struck by Finger 1, which is the thumb. Over the course of the play, the Index, Middle, and Ring fingers rest lightly on what may be called their "home" keys , , and , or hover just above them, and of course depart from them whenever called upon to strike some key either higher or lower in the same column, but return to them — to the home keys — whenever convenient. The bump which many keyboards have on the  key is felt by the middle finger to give reassurance of correct finger placement without having to look.
If these fingering rules are adhered to, the player will soon be able to control ROBO rapidly and reflexively, and without having to take his eyes off the action. If these fingering rules are neglected, the player will waste time looking back and forth between PlayScreen and keypad, which will degrade the quality of his play, and handicap him in competition.
Below are examples of how each of the six controls might be used, for the time being assuming that each KeyStroke rotates ROBO 45°. On the upper-right of each example is shown a single letter, as for example the R just below (which happens to be lying on a blue background, and tipped 90° CounterClockWise) — this is the MandatedView, MandVu for short, which the player will enable ROBO to see by rotating him.
PITCH UP AND PITCH DOWN
The PITCH command is the easiest to learn. Although the task is to orient ROBO so that he sees exactly the MandVu that is shown on the ScoreBoard, in Etude 1 this can be achieved merely by laying ROBO face down parallel to the cube wall, by means of a single key tap. Although a player who knows (s)he is playing Etude 1 is able to achieve ROBO MandVu merely by thus flattening him, and thus paying no attention to what is displayed on the wall, in more advanced Etudes, it will be necessary to pay attention to the Character on the Wall, and therefore doing so from the outset will pay off in the long run.
||To enjoy this Mandated View (MandVu) of a BlueWall R, ROBO needs to PITCH UP 45°, so with the index finger of his right hand, the player taps .
||To enjoy this MandVu of a PurpleWall T, ROBO needs to PITCH DOWN 45°, so with the index finger of his right hand, the player taps .
The three small ROBOS are memories of ROBO wall conquests earlier in the game. Incidentally, it is the about-to-achieve-MandVu ROBO in the ScreenCapture above who appears as a miniature memory-trace in the ScreenCapture below, enjoying the MandVu.
YAW RIGHT AND YAW LEFT
The YAW command requires a new skill to be learned, which is identifying ROBO's right and left hands, which is not always easy, given the strange orientations in which he often appears. At the same time, if, for example, playing Etude 5, which requires only a single YAW KeyStroke, the player will be able to get the correct answer while continuing to pay no attention to the orientation of the character on the wall. Simply placing ROBO's ventral surface parallel to the wall accomplishes all that is required.
||To enjoy this MandVu of a GreenWall E, ROBO needs to YAW RIGHT 45°, so with the middle finger of his right hand, the player taps .
The smaller of the two ROBOs below is a memory of the red wall that ROBO conquered earlier in the game.
||To enjoy this MandVu of a RedWall P, ROBO needs to YAW LEFT 45°, so with the middle finger of his right hand, the player taps .
ROLL RIGHT AND ROLL LEFT
The ROLL command is the most difficult of the three because it requires not only that the player identify ROBO's right and left hands, but for the first time necessitates that he examine the character that is painted on the wall, in order to decide in which direction ROBO needs to be rotated to achieve MandVu. Below are examples from Etude 9, where a single tap of either  or  produces the requisite 45-degree ROLL RIGHT or ROLL LEFT.
||To enjoy this MandVu of a YellowWall T, ROBO needs to ROLL RIGHT 45°, so with the ring finger of his right hand, the player taps .
The smaller of the two ROBOs visible below is a memory of his conquest of the blue wall earlier in the game.
||To enjoy this MandVu of a BlueWall R, ROBO needs to ROLL LEFT 45°, so with the ring finger of his right hand, the player taps . Incidentally, the ROBO below who is about to achieve MandVu is shown above as a memory-trace which has achieved that very same MandVu (the above image having been ScreenCaptured later in the same game).
Actually, the above six examples do not typify what is most usually seen during play. During play, ROBO is often on the far side of the cube, where he, and the character he has to deal with, would be hidden behind such opaque walls as are shown above, making it impossible for the player to know how to proceed. It is for this reason that the default settings under CUBE APPEARANCE in the ControlPanel ScreenCapture below tend toward cube transparency. To achieve the high opacity evident in the six ScreenCaptures immediately above, the player would have to implement wall-appearance options 30 and 34 on the ControlPanel, which he is unlikely to want to do.
At the moment, the SpaceCraft program is available only for Windows.
SpaceCraft should be played on a computer connected to the largest and highest-resolution screen available, and having a keyboard that has a separate numerical keypad on the right.
twelvebytwelve.net/spacecraft/spacecraft-012.exe, download spacecraft-012.exe, which possibly will automatically be placed into your Downloads folder.
Create a folder titled spacecraft anywhere you want on your computer, and move spacecraft-012.exe into the spacecraft folder. That way, the data file which stores your performance stats will be created in the same folder as spacecraft-012.exe, instead of in some remote default location, like on your desktop. If the PlayerName you select for yourself is, say, CINDY, then your performance statistics will be stored in the folder/file: spacecraft/player-cindy.txt.
If you download this SpaceCraft User Manual into the same spacecraft folder, and if you open your copy of the manual by double-clicking on it in your directory, then you should be able to get the SpaceCraft program running merely by clicking here.
The first screen to appear asks for a PlayerName. The PlayerName makes it possible for SpaceCraft to record your performance statistics, and to display summaries of them during play.
Shown below is a user having typed an unacceptable PlayerName four times, before finally typing one that was acceptable — taras.
On the first attempt, the player hit [ENTER] (which on some keyboards is designated the [RETURN] key) before having typed in any characters at all. Next came a PlayerName too long. Then one containing an ampersand, &, which is not either an alphabetic or a numeric character. Then one containing a space, which also is neither an alphabetic nor a numeric character.
The PlayerName taras, is acceptable, and if the player whose work is shown below now hits [ENTER], then TARAS will be the name under which his performance data will be stored. The case, upper or lower, of alphabetic characters doesn't matter, as SpaceCraft will capitalize them all upon receipt. Thus, SpaceCraft will assume that TARAS and Taras and taras, and so on, all refer to the single player TARAS. More generally, the effect in SpaceCraft of any KeyStroke is never altered by [SHIFT] or [CAPS LOCK].
Where only a small number of players is expected, and small possibility of duplication, a PlayerName as short as a single character may be sufficient, say T (which is the same as t). As the number of players increases, confusion will be avoided by using longer PlayerNames, and ones that can be expected to be unique, as for example taras180, or even more complicated, as needed.
After you have typed a satisfactory PlayerName, hitting [ENTER] takes you to the ControlPanel.
Don't be bothered by the seeming complexity of the ControlPanel shown below — you will soon enough come to realize that it is extremely simple. All you need to notice at the moment is the very top and the very bottom.
At the very top of the ControlPanel is the PlayerName TARAS, its alphabetic characters having been capitalized, as expected. The zero to the left of the PlayerName indicates that hitting [ENTER] will return to the PlayerName page where a different PlayerName can be entered.
And at the very bottom of the ControlPanel appears a choice of two ways of escaping the ControlPanel — either to play SpaceCraft, or else to exit SpaceCraft altogether.
So, if TARAS wishes to begin play without making any changes to the control panel, he hits the three keys [ENTER], which can be expressed more succinctly as [ENTER]. Opposite the question mark at the bottom of the ControlPanel, TARAS can be seen to have already typed , and is presumably about to hit [ENTER] to begin playing.
Had TARAS instead wanted to exit SpaceCraft, he would have typed [ENTER].
Maybe it would be a good idea for you to yourself begin playing by hitting [ENTER] right now. You will be presented with the PlayScreen where you will need to Pitch ROBO 45° either Up or Down, whichever succeeds in flattening him against the nearest wall of the cube, and which flattening you can accomplish by tapping either  or  on your numerical keypad with your right-hand forefinger (same as index finger, or if the thumb is considered to be finger 1, as it is to piano students, then it is finger 2 that is to be used at the moment). What you will be playing is Etude 1, and your goal should be to keep on practicing it until you can play it without error ten times in a row. The goal should be not to play the Etude as fast as possible, but rather to be able to play it flawlessly every time.
Each time you conquer the six sides of the cube, you tap  to come back to the ControlPanel, then [ENTER] to play the same Etude again, and what you will be playing initially is Etude 1. It is after you have been able to play Etude 1 without error ten times in a row that you will understand SpaceCraft well enough to go on to Etude 2.
Just below the PlayerName section on the ControlPanel can be seen the choice of Studies (or Etudes as they are called here) that you need to practice. That "1 Pitch45" is colored indicates that Etude 1 is being offered by default, the Pitch45 explaining that Etude 1 offers practice on the Pitch rotation through 45°. PYR signifies practice on all three rotations — Pitch, Yawn, or Roll.
When the day arrives that you want to move on to Etude 2, say, in the ControlPanel you will hit [ENTER], whereupon color will shift from "1 Pitch45" to "2 Pitch90", following which you can start playing Etude 2 by hitting [ENTER]. Etude 2 is the same as Etude 1, except that ROBO needs to be rotated 90° instead of 45°, and so as each KeyStroke moves ROBO only 45°, you are going to have to hit the correct key twice. Similarly, when you get to Etude 3 where a 135° rotation is required, you are going to have to hit the correct key three times.
The 16 Etudes being offered in the current version of SpaceCraft represent an introductory level of play. More advanced play is available in expanded versions.
No harm in experimenting with changing the Etude setting a few times right now, though it is advisable to play only Etude 1 until smooth and confident performance is achieved. Generally speaking, learning will proceed most expeditiously if lower-level Etudes are fully mastered before moving on to higher-level ones.
If you do initiate play on the PlayScreen by means of [ENTER], but for whatever reason want to return to the ControlPanel before completing play, just tap , which, however, credits you with an INCOMPLETE play, which for purposes of computing %WON is equivalent to a LOSS. Best practice would be to never hit , that is not until all six sides of the cube have been conquered. If you make a mistake early in play, finish the play with great care even though it will be counted as LOST, and don't consider yourself to be wasting your time, because the extra practice builds skill whether it wins you a personal best speed or not.
On the PlayScreen, a KeyStroke does not have to be followed by [ENTER] for SpaceCraft to respond to it, which is why we give PlayScreen KeyStrokes a different name, "tap". On the ControlPanel, then, we "hit" a key, meaning that SpaceCraft will respond to what we have typed only after we hit [ENTER], but on the PlayScreen, we "tap" a key, meaning that SpaceCraft will respond to that KeyStroke without our hitting [ENTER]. In fact, [ENTER] is needed only on the ControlPanel; if tapped on the PlayScreen, it is ignored.
Each side of the rotating cube visible during play is considered to consist of a WALL, which serves as a background or canvas on which a CHARACTER is drawn. Glancing through the cube portrayals below will reveal that the WALL may be colored, or transparent, where a transparent wall could be thought of as a glass wall. And that the CHARACTER, in turn, may be colored, or white (which for our purposes is not considered to be a color), or transparent. A transparent CHARACTER on a transparent WALL signifies that the entire cube face is transparent, except for the visible outline of the character.
The default choices showing in the ControlPanel above will possibly be acceptable for most purposes, although testing other combinations may lead to an improved experience. What needs to be balanced are two clashing needs: on the one hand, the more opaque the side of a cube, the more clearly can its orientation and direction of motion be understood; on the other hand, the more transparent a side, the more clearly can a ROBO behind it be seen. It is in the attempt to maximize these two opposing needs that appearance shifts upon ROBO's arrival at a face — but this is a phenomenon that needs to be experienced to be understood, and need not concern the player in his initial introduction to SpaceCraft.
Still lower down on the Control Panel, the ECHO section controls sound, and more specifically sound that reflects what the player has just accomplished, and in this sense the sound which echoes what he has just done.
ECHO=40 responds to RoboControl key taps with silence.
ECHO=41 is the default — after tapping any of the six RoboControl keys, (the digits  to ), a voice describes what that key-tap accomplished, the "Pitch Up" showing opposite ECHO-41 in the ControlPanel ScreenCapture above serving as an example. Other examples would be the voice saying "Pitch Down" or "Yaw Left" or "Roll Right".
ECHO=42 introduces a sinking sound (referred to more briefly as the SUNK sound, and referred to most briefly of all as just plain SUNK) which follows a player tapping the final key which orients ROBO correctly. For example, if three KeyStrokes are required to bring ROBO to MandVu, the first two will be followed by an audible "Yaw Right", say, and the last one will be followed by the SUNK sound, by way of confirming that the last KeyStroke was the culminating step of the correct solution. Imagine a golf player who takes several swings to bring his ball near the hole, and recognizes that his final putt is the one that SUNK it.
ECHO=43 produces no sound following any RoboControl KeyStrokes prior to the final one, and where the final one produces SUNK.
PAST PERFORMANCE REACH
A summary of a player's past performance is on display during play, as will be described further below, how far back into the past that summary reaches being set by default to the most recent ten performances, as can be seen in the ControlPanel display above. A player wishing to see only his last 5 performances summarized would hit [ENTER], then respond to the question "PastPerformanceReach ?" by typing [ENTER]. A player wishing to see his past 20 performances summarized would type [ENTER], then [ENTER]. When PastPerformanceReach exceeds the number of performances that has been recorded, then the number summarized will be the total number that has been recorded; therefore a player wishing to see a summary of all past performances could set PastPerformanceReach to a number well beyond the number of past performaces already recorded, or likely to ever by recorded, as for example 10000.
KEYSTROKES THAT PRODUCE AN EFFECT DURING PLAY
Upon first hitting [ENTER] and encountering the rotating cube, the player will be able to observe the effects of tapping the following keys (if get no response, it's possibly because the focus has fallen outside the PlayScreen, the solution to which is bringing the mouse cursor inside the PlayScreen, and there clicking the mouse, which should restore responsivity to the KeyStrokes under discussion here):
[E] stands for Echo, which is presented here first because we have just been talking about controlling sound using the ControlPanel. Tapping [E] during play, then, runs sound in a cycle through the ECHO=40 TO ECHO=43 alternatives that we have just taken a look at above, starting at the default ECHO=41. The change in what is being echoed will not become audible upon simply tapping [E], but becomes audible, rather, upon the player's subsequently tapping the RoboControl keys  to .
[ESC] exits the SpaceCraft program entirely. To restart SpaceCraft, you will need to retrieve your mouse and double-click on spacecraft-012.exe
[~] is the tilde which lies to the left of the  key that is on the upper-left of the KeyBoard, and whose effect is to freeze all motion. Tap [ENTER] to unfreeze. Useful during learning to permit unhurried study of a situation that in the heat of play seemed suddenly puzzling. The same freezing can be accomplished by tapping [F].
With the cube rotating on the PlayScreen, go ahead and try tapping [~], and all the other keys discussed in this section, and as often as you like, to familiarize yourself with their effect.
[W] stands for Whither Wander? and changes the direction of ongoing motion to a random new direction. Is useful occasionally, as when a cube face is seen from the side, making it difficult or impossible to see what's on it, and so when it needs to be attracted toward the center of the PlayScreen where it will face the player. Several taps of the [W] key will often be needed to get rotation in a new direction close to the one desired.
[S] stands for Starry Sky. Tapping [S] repeatedly takes the player in a circuit of the nine available backgrounds. May be used simply to provide the most agreeable background. Also, useful to test ability to perform under distraction. The stationary stars get very slowly bigger over time, which contributes to a 3D effect. However, the denser backgrounds may be so demanding of computer resources that they slow performance.
[X] produces a one-second X hatching on the top of ROBO's head, along with a striping of the face front and sides. Helps the player clarify what ROBO orientation is, which is least obvious when ROBO is being seen on edge. The ScreenCapture image below shows striping on the right side of ROBO's face, in the present case admittedly unhelpful because ROBO's orientation happens to be already manifest. To lessen ambiguity in the sometimes small and unclear ScreenCaptures shown further below, ROBO's head is often shown with this head-locating facilitator on.
[D] stands for Droit, which in French means right-hand, tapping which helps the player identify ROBO's right hand by lighting it up, as shown in the CAPTURE image above, but only for one second. As being able to quickly and unerringly identify ROBO's right and left hands, no matter what ROBO's orientation, is one of the fundamental skills that needs to be acquired, calling on [D] to make the identification should be relied upon rarely, and only to confirm an identification that the player has already decided.
 stands for ONCE MORE, and takes the player back to the control panel. To replay the same Etude as is currently playing, tap  while on the PlayScreen, then hit [ENTER] on the ControlPanel.
A slightly more complicated example. Say that after completing Etude 1 (or in the middle of playing Etude 1) you want to play Etude 2. Your KeyStrokes would be   [ENTER]  [ENTER], where the  is tapped on the PlayScreen, and the others are hit on the ControlPanel. That's six KeyStrokes, which with practice can be accomplished within a couple of seconds, and without having to look down at the keyboard, for anyone who has learned to touch-type on the numeric keypad, which requires adherence to the fingering specified in the keyboard diagram above, where gray numbers 1 to 5 have been added to the keys to indicate which right-hand fingers must be used to type which keys, where the thumb is considered to be Finger 1, and which Finger 1 the above diagram indicates is allocated sole responsibility for striking the [ZERO] key.
Remaining to be discussed are keys usable only during the PostPlay EXPLORATION interval: [A] and [Z],
as well as the six keys [R] [G] [B] [Y] [P] [C] standing for Red Green Blue Yellow Purple Cyan, and which keys can be seen in the keyboard diagram above to be highlighted in those colors, and which keys, during PostPlay EXPLORATION, control the cube faces bearing those six colors. More on these additional eight keys, then, under the heading POSTPLAY EXPLORATION below.
On the upper-left of the play screen appear six jumbled characters spelling a word on backgrounds whose colors are RGBYPC, and which characters can be seen to spell out the same word on the large rotating cube as well, though not jumbled in the same way as on the ScoreBoard. ROBO always starts off positioned opposite the red wall of the cube, the wall initially facing the player, and as play on each wall finishes (as ROBO "conquers the wall", so to speak), ROBO automatically moves to the next wall (following the order RGBYPC) to attempt to conquer it as well.
The players task is to orient ROBO so that ROBO sees the character on the wall before him the way it appears on the ScoreBoard, which will be called here the MANDATED VIEW, abbreviated to MandVu. Upon the player orienting ROBO into the MandVu, the CUBE will go into a SHUDDER reflecting that the CUBE resists attemts at conquest (because that's what ROBO is doing — removing the walls so that he can get at the GEM rotating in the middle, but more about the story line later). Over the course of the SHUDDER, ROBO shrinks in size to what may be thought of as a memory-trace of himself as he was at the moment of reaching MandVu orientation. A memory-trace ROBO will be left hovering over each conquered wall until the end of the game.
Once the cube's SHUDDER with ROBO shrinkage have been completed, ROBO appears full size opposite the next wall he needs to conquer (from red wall to green wall will be the first such transportation).
The ScreenCapture below shows a typical ScoreBoard at the end of a game. The letters on the six sides of the cube happen to spell out REBUKE. The numbers in the upper row indicate the minimul number of KeyStrokes needed to reach MandVu, and the numbers in the lower row indicate the number of KeyStrokes the player actually expended. The two rows matching indicates that the player played a perfect game, never making more KeyStrokes than were necessary. The player's time was 99 seconds, which the program accepted for entry into the players performance log because the player made no mistakes, as evidenced by his having made zero moves beyond those necessary.
The ScoreBoard below shows a different outcome. The word shown is FEEDER. To conquer the green E took the player 6 Keystrokes instead of the allocated 4, two KeyStrokes beyond what was necessary. Whenever any excess moves are made, performance time is not credited, which is why the 70 seconds showing is "rejected". Accuracy is more important than speed. If performance has been error-free, the performance is always recorded, no matter how slow, and if the performance is either the player's first performance of the given Etude, or is faster than any previous performance of that Etude, then the player is credited with a PersonalBest. The daily accumulation of PersonalBests produces prodigious learning as the days fly by, and the infinitesimal improvement that is needed to be counted as a PersonalBest guarantees that the accumulation will be stress free.
The ScoreBoard below shows perfect performance over the first three letters of EFFETE, but for the fourth letter, 9 KeyStrokes expended so far, when only 1 was needed. A curious apparition here is that the ScoreBoard orientation of the yellow E differs between upper and lower rows. What the upper row shows is the appearance of the E that is required (the MandVu); what the lower row shows is the appearance of the E that ROBO at the moment has attained. In other words, despite having expended 9 KeyStrokes, ROBO has not yet achieved MandVu. Generally, before achieving MandVu, the lower row remains blank, and upon achieving MandVu, the lower row shows the number of KeyStrokes used to attain MandVu. The new phenomenon illustrated below is that the lower row of icons also shows what ROBO sees when his face is parallel to the wall, but when he is ROLLED either 90 or 180° off optimal placement.
PAST PERFORMANCE STATS
Below is one version of performance stats which are projected onto the upper-right of the PlayScreen.  PlayerName happens to be TARAS. That PastPerformanceReach equalling 10 limits the summary to only the most recent 10 performances can be inferred by noting that the sum of WON + LOST + INCOMPLETE never exceeds 10.
SEC shows TARAS's personal-best time in seconds to complete play (among the most recent 10 past performances). KS is the number of KeyStrokes that was needed to achieve that particular personal best, where KS remains constant at either 6, 12, or 18, except during Etudes 4, 8, 12, and 16, where the number of KeyStrokes can vary because the "any" option chooses the degrees of rotation randomly for each of the six sides of the cube.
A play is considered WON if completed using the optimal number of KeyStrokes, and is considered LOST if completed using more than the optimal number. INC records the number of plays that were started but abandoned before all six sides of the cube had been conquered, which is to say were left INCOMPLETE, which most usually happens when a player makes a mistake in mid-play, and prefers to start a new Etude without completing the one he is currently playing. A player is considered to have started play when he correctly orients ROBO on the first face of the cube, which is the red face.
%WON is the percent of games played that were WON, where games played includes those WON plus those LOST plus those left INCOMPLETE. %WON is the statistic that the player should work on incrementing to 100%, and should strive to keep at 100% during further practice, rather than striving to increase speed so as to lower his personal best SEC.
A row of Asterisks (not shown) would indicate no recorded successful completions (equilvalent to WON equalling zero), resulting either from no attempts having been made or else attempts but not a single one of them WON; however, in the case of asterisks, any number of LOST or INC could have been played, but will not appear in the performance display unless they can be accompanied by at least a single WON.
The colored line reminds Taras that he is currently playing Etude 5, where a single KeyStroke productive of Yaw ±45° is needed to put ROBO into MandVu. The colored balls are stars in the permanent firmament.
Some of these particular performance statistics are implausibly regular, and perhaps implausibly poor, because they were generated by Taras playing through many of the alternatives searching for program bugs rather than to improve his performance.
One particular feature of this chart should be noted, and is worth emulating. It is that Taras has begun to advance through the Etudes methodically earning 100% WON from Etude 1 to Etude 4. The chart shows that all of these were accomplished late in the evening of 22Aug2015, within only a few minutes of one another, with the time recording when the personal best among the ten was achieved.
Such, then, is the path that should be followed by anyone wishing to achieve mastery of SpaceCraft — start at Etude 1, and stick with it until reaching 100% WON, and only then go on to the next Etude, and so on.
The older the date showing on a chart, the more likely that a repeat play is needed to strengthen a fading skill. The chart, then, lays out a path of optimal practice. And even when the day arrives that the chart shows every Etude from 1 to 16 accompanied by a 100% WON, weaknesses calling for further practice will be identified by an unusually long time appearing in the SEC column, or a particularly stale date appearing in the DATE column.
Once these sixteen Etudes are mastered, it will be time to move on to MandVus requiring more than one of the six possible rotations, as for example a PitchUp45 followed by a YawRight90.
After ROBO has conquered all six walls of the cube, the cube continues to rotate with a small, memory-trace ROBO, continuing to enjoy his MandVu, attached to each cube face. Although the six RoboControl keys  to  have stopped working, as has [E] for ECHO, all remaining PlayScreen controls do continue to work, namely
[~] to freeze motion, followed by [ENTER] to unfreeze
[W] for Whither Wander, to randomly change direction of cube rotation
[S] for Starry Sky to change the heavenly background
[X] for the X hatching on ROBO's scalp, and the lining of his face, to facilitate one-second head locating
[D] for the French Droit, which for one second brightens ROBO's right hand
And of course [ESC] still continues to Exit the program, and  continues to take the player to the ControlPanel where he can request, among other things, to play another Etude, whether the same as he has just finished playing, or different.
In addition, though, a number of keys that had remained inactive during PLAY become activated during post-play EXPLORATION:
[Z] stands for near-Zero obstruction of viewing. That is, [Z] strips all six faces of the cube down to merely the outline of its Character, which overall effect could have been implemented throughout PLAY by earlier selecting WAVE APPEARANCE 33 and 37 on the ControlPanel.
[A] standing for All six faces of the cube being simultaneously circulated through the following five appearances:
(1) solid colored wall with nothing written on it,
(2) character appears in white (same as ControlPanel 30 and 34),
(3) character is turned into transparent glass (ControlPanel 31 and 35),
(4) wall disappears, leaving character in solid color (ControlPanel 32 and 36), and lastly
(5) only the outline of the letter (the state that is produced on all walls by hitting [Z], which is equivalent to implementing ControlPanel 33 and 37).
If the walls are unequal in transparency to begin with, hitting [A] will not impose equality — it will cycle each cube face through its five appearances independently of what's happening on the other five cube faces.
And, to crown all, repeatedly hitting one of the keys [R] [G] [B] [Y] [P] [C] (standing for RED GREEN BLUE YELLOW PURPLE CYAN) cycles the wall typified by that color through the same appearances that hitting [A] took all the walls through. In competitive, two-person play, is used by a player to erect obstacles to the opponent's clear view of what's happening.
By way of illustration, we see below the ScoreBoard after playing Etude 20, featuring the word LAMBDA. The KeyStroke numbers in the lower row matching those in the upper row indicates that the player completed the game without error. The time of 86 seconds is long because the player was pondering and debating, and not at all trying to set a speed record.
As the cube continues to rotate after game conclusion, the player hitting [Z] (which, it may be recalled, offers near-Zero obstruction to viewing on all six cube sides) creates a view like that below, which (even here, despite its low resolution) permits the skill-building exercise of verifying that each memory-trace ROBO is indeed enjoying the wall's MandVu which we saw stipulated above:
And as the cube continues to rotate during the post-play EXPLORATION interval, the player is able to construct walls having different degrees of transparency, as shown below:
In addition to enjoying such mobile displays during PostPlay EXPLORATION because they activate powerful, but sometimes dormant, mental skills, and because they activate the love of building, three exercises during PostPlay EXPLORATION will contribute toward reinforcing and expanding spatial skills:
(1) Proceeding through the color sequence RGBYPC, read the hidden word, both on the ScoreBoard and on the Cube, which in the three ScreenCaptures immediately above can be seen to be LAMBDA.
(2) Go from one memory-trace ROBO to another, in random order, and for each visualize what his character looks like to him, and then confirm your vision by consulting the ScoreBoard MandVu.
(3) Go from one memory-trace Robo to another, in random order, and for each decide which is his right hand, and only when you are confident of your decision, verify by tapping [D]. This is by far the most difficult of the three exercises and so the one most worth practicing to mastery — to the point where you can identify every ROBO's right hand as quickly and unerringly as you can identify his head.
UNUSED STORYLINE FEATURES
The gem rotating independently in the middle of the cube is part of a storyline which at the moment remains undeveloped.
Another feature not as yet integrated into a storyline is that for every KeyStroke beyond the number alloted for the solution of any face of a cube, ROBO loses a random one of his 69 surface plates. A player who wants to see this loss progress to a large number can keep on tapping a rotation-control key which is incapable of arriving at the right answer. For example, during Etude 1 (which requires one of the Pitch-control keys, either  for Pitch Up or  for Pitch Down) keep tapping only  which keeps producing only an unhelpful Yaw Left. As the number of taps approaches 69 more than had been allocated, ROBO will be seen to approach complete disappearance.