It's great to see all this straightedge and compass stuff turned into a game, and I absolutely applaud the developers for launching it.
The above math, plus some graph theory (biconnected components etc.) is what drove D-Cubed's DCM 2D component dated, which powered the sketcher portion of many parametric MCAD systems after D-Cubed's launch by John Owen in 1989. See his paper "Algebraic Solution for Geometry from Dimensional Constraints" referenced on the Siemens PLM site:
The original parametric CAD system, PTC's Pro Engineer (now Creo) predates that (1987) and had it's own numeric (Newton-Raphson) solver. John Owen's innovation of using Galois Theory combined with Graph Theory to solve straightedge and compass configurations was a significant technical advance at the time, and the DCM 2D component ended up powering most sketchers in the industry.
Cool! I've never seen a practical example using Galois theory before. 10 years of computer graphics experience, but very little maths beyond my physics degree...
I enjoy the game a lot, but I disagree that it has that much of educational value. Without pre-existing geometry knowledge (e.g. similar triangles; median, bisector, and altitude in an isosceles triangle) the way to solve puzzles is only to randomly screw around. The game then doesn't give any explanation why that solution works.
After randomly screw around I usually wonder why my solution worked. If I didn't do that it will be much harder to solve the next, more challenging level. Now I'm playing on beta level packs (just one pack after the first one) and it takes time to solve the puzzle if I only randomly screw around.
I have been using this to teach my two daughters (10 and 14) geometry. Would love to find more sites like this that make teaching maths fun. I do, however, give them hints on how to solve the puzzles and explain the theory of how it works.
This is the greatest game I've found this year. I've been craving this. The way geometry and trigonometry was explored a long time ago by playing with arcs and lines is just so so cool to me.
Definitely agree. From what I can tell so far (I've only just unlocked the Perpendicular Bisector tool), it strikes a great balance. Just-in-time helper prompts, a nice drag 'n' drop style UX that naturally lends itself to mobile, and clear instructions.
I like that it uses the concept of Photoshop-style "tools" for drawing points, segments, circles, etc. A very natural digital extension of the physical pencil, straight edge, and compass, with some nice abstractions built on top.
I appreciate that the app is free, with the option to pay to unlock all levels without the burden of completing the previous ones. That feels like a very accommodating model and gives users a reason to pay without creating content exclusively for paying users.
It's a little unfortunate that they don't accept my somewhat elaborate but ultimately correct solutions, if I find a way to create a perfect rhombus within a rectangle in Lwaytoomany Ealsowaytoomany, they should accept it at least, and give me 1 star for my effort!
Pretty fun so far though!
edit: So far, I've found two methods to find the center of a circle that they don't like. 5L E7 isn't that bad
As far as I can tell, solutions are recognized as long as they are valid, regardless of length. Can you post an example of a solution you feel is correct but that isn't being accepted?
Here[1] is one that's not being accepted. Here I have to draw an equilateral triangle given a straight line.
I am drawing two circles, on the two terminal points, with radius as the length of the given line. The intersection of two circles is the third point of the triangle.
In principle, this is the right construction. However, you did not perform a proper construction, as can be seen by the two tiny red circles: Instead of constructing a circle passing through the end points of the given line, you "eyeballed" the circle diameter. Which may look right at this resolution, but if you were to zoom in sufficiently, you would inevitable end up with a circle that oh-so-slightly misses to go through one of the end points of the base line.
In the smartphone app, to do this right you need to select the circle tool, start at one end point of the line, then drag your finger exactly to the other end point -- it'll "snap" to that, and create a circle through the second end point.
Yes, I thought at too so I zoomed in to make sure that I am not making that mistake.
I've found a solution though. The second point that indicates the radius of the circle also snaps to other points. I didn't know that.
So now I can start dragging from one end point of the given line and end on the other point(not eye ball it) to make sure the circle is the exactly that radius.
Solutions are accepted, no matter how many moves. It sounds to me you may be eyeing things. You need to start from a predefined point and end at another predefined point, you can't just end at random places, it won't give you the solution then.
I've played this game and found solutions that I really believe are correct but aren't accepted, and need to redo the solution another way. The game is certainly not perfect.
But I'd say overall it's a great game. It's been my commute puzzler for a while now. Trying to get optimal solutions can get really tricky.
I've played this a lot and introduced others to it. When others make the claim you do here I have always found that their solutions look good and plausible, but are not exact.
It may be that you are the exception, and that your constructions are correct, but as with compiler errors, the problem is statistically more likely to be in your construction.
I'm sure the devs would love to see a construction that is proven to be correct, but is not recognised by the system. Having a proof is key.
Keeping in mind that I have absolutely no idea what I'm doing, was never taught any of this stuff at school, and am stone cold stuck on 1.5, why is this not a solution to the Rhombus in Rectangle problem? It seems to match perfectly to the diagram and doesn't have any red points.
It's impossible to tell exactly what you've done, but here's what it looks like.
Using the corner at bottom left as centre you've drawn a circle with radius equal to the long side, then you've done the same with the corner at bottom right. That gives you the point above, but closest to, the rectangle. Then you've joined that point to the corner at bottom left and marked the point where it crosses the top line.
There is no reason why the length of the line crossing the rectangle should be equal to the right section of the divided top line. The top line of the rectangle can be moved up and down, and the two line lengths will vary in opposite directions. At some point they'll be equal, but in general they won't, whereas the problem requires that all four sides of the rhombus have equal length.
If that's not clear I can write a longer description with diagrams, but I hope that explains why what you've got isn't a solution, even though it looks good.
I was going to reply to you off-line, but you have no contact details in your profile, despite saying "Get in touch!"
Perhaps you didn't realise that the "email" field in your profile is private and only visible to the mods. Logout and have a look at your profile and you'll see what I mean - no contact details.
Hope you see this! Personally, I use "HN Replies"[0] to let me know if/when people reply to my comments, just for cases like this.
That construction doesn't have 4 sides of equal length (although it's really close), which is required for the solution (you can see that by the 4 sides having a single mark through them, that indicates they have the same length)
In addition to what the other commenters have said, you can press the "solution" button to see what the final rhombus should look like, and whether yours falls exactly on it.
Follow-up (the app I use on mobile doesn't have an edit button) - it would be interesting to provide a section that gives impossible challenges and declares them as such, such as constructing a heptagon and squaring the circle, along with an explanation of why they are impossible. It might ignite more common interest in Galois theory!
OTOH, it might be more useful to show people what can be geometrically constructed. Since Ancient Greeks knew that circles CAN BE squared, regular heptagons CAN BE constructed, and angles CAN BE trisected by neusis methods[1]. I don't think this is widely taught. Why don't we teach what can be done rather than what can't be done when you limit yourself to straight edge and compass?
I wonder how difficult it would be to add neusis to the construction techniques in Euclidea or GeoGebra.
It would be nice to be able to move or get rid of the task completed box to review your solution and look for modifications. Then you could reset and replay. The inability to review after completing is annoying.
There's a formula for the coördinates of where two arcs meet, where a line and a circle meet, etc., in terms of the coördinates of the points which generated the lines and circles — and you only need a few such formulæ (linear and quadratic equations, in fact) to cover all legal straight-edge-and-compass constructions.
So keeping track of exact, algebraic locations of the points is basically equivalent to manipulating exact expressions for roots of polynomials, which is not totally trivial (the tricky bit is simplifying roots of a quadratic equation whose coefficients were given as roots of a previous quadratic equation) but it's the sort of thing Maple and Mathematica have been doing for decades.
The above math, plus some graph theory (biconnected components etc.) is what drove D-Cubed's DCM 2D component dated, which powered the sketcher portion of many parametric MCAD systems after D-Cubed's launch by John Owen in 1989. See his paper "Algebraic Solution for Geometry from Dimensional Constraints" referenced on the Siemens PLM site:
https://www.plm.automation.siemens.com/en_us/products/open/d...
The original parametric CAD system, PTC's Pro Engineer (now Creo) predates that (1987) and had it's own numeric (Newton-Raphson) solver. John Owen's innovation of using Galois Theory combined with Graph Theory to solve straightedge and compass configurations was a significant technical advance at the time, and the DCM 2D component ended up powering most sketchers in the industry.
Disclaimer: I worked at D-Cubed 1995-2000.