//	Program:
//		4D Draw
//
//	Language:
//		English
//		English
//
//	Translator:
//		N/A
//
//	Date:
//		N/A
//
//	© 2025 by Jeff Weeks
//	See TermsOfUse.txt

//	Touch modes (text currently unused, for lack of space)
"Neutral"		= "∅";
"MovePoints"	= "Move Points";
"AddPoints"		= "Add Points";
"AddEdges"		= "Add Edges";
"DeletePoints"	= "Delete Points";
"DeleteEdges"	= "Delete Edges";

//	Options
"Box"			= "Box";
"SnapToGrid"	= "Snap-to-grid";
"Inertia"		= "Inertia";
"Coordinates"	= "Coordinates";

//	Rotate
"Rotate %@"		= "Rotate %@";

//	Lock/unlock figure content
"LockFigure"	= "Lock %@";
"UnlockFigure"	= "Unlock %@";

//	Help - How to Draw

"How to draw" = "How to draw";

"Intro-what" = "4D Draw";
"Intro-how" = "is a simple vector graphics app that lets you draw 4‑dimensional figures. It represents each point’s first three coordinates in the usual way, while using color to represent the fourth coordinate.";

"AddPoints-what" = "To create new points…";
"AddPoints-how" = "tap the Add Points button, then drag from the walls to create new points.";

"AddEdges-what" = "To create new edges…";
"AddEdges-how" = "tap the Add Edges button, then tap the points that you want to connect.";

"MovePoints-what" = "To move existing points…";
"MovePoints-how" = "tap the Move Points button, then drag points to move them. Drag a point along its rainbow-colored w‑axis to move it in the fourth dimension.";
"MovePoints-note" = "If any two of a point’s four axes happen to align on your 2‑dimen\U00ADsional display, you may rotate the whole scene to separate them.";

"DeletePoints-what" = "To delete points…";
"DeletePoints-how" = "tap the Delete Points button, then tap the points you want to delete.";

"DeleteEdges-what" = "To delete edges…";
"DeleteEdges-how" = "tap the Delete Edges button, then tap the edges you want to delete.";

"RotateScene-what" = "To rotate the whole scene…";
"RotateScene-how" = "drag the background (in any mode except Add Points).";

//	Help - Construction Ideas

//	Note:  In most languages other than English,
//	title capitalization is the same as sentence capitalization.
"Construction ideas" = "Construction ideas";	//	sentence capitalization for use in menu
"Construction Ideas" = "Construction Ideas";	//	title capitalization for use in panel

"5-cell title" = "Construct the 4D analog\nof a tetrahedron";
"5-cell text" = "Arrange four red vertices symmetri\U00ADcally around a central violet vertex. The color shows that the red vertices sit in the negative direction in the fourth dimension (w < 0) while the violet vertex sits in the positive direction in the fourth dimension (w > 0). The hypertetrahedron's five 0D vertices and ten 1D edges will be obvious. Can you also find its five 3D hyperfaces (“cells”)? More challenging still, can you find all ten of its 2D faces?";

"16-cell title" = "Construct the 4D analog\nof an octahedron";
"16-cell text 1" = "Hint: An ordinary 3D octahedron may be drawn with its vertices at (±1, 0, 0), (0, ±1, 0) and (0, 0, ±1). Similarly, a 4D “hyperoctahedron” may be drawn with its vertices at (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0) and (0, 0, 0, ±1).";
"16-cell text 2" = "Your hyperoctahedron's 0D vertices and 1D edges will be obvious. Can you also find its 2D faces and 3D cells?";

"8-cell title" = "Construct the 4D analog\nof a cube";
"8-cell text 1" = "You'll need eight red vertices and eight violet vertices. Each red vertex sits at exactly the same 3D location as the corresponding violet vertex (same x, y and z coordinates). Which pairs of vertices do you need to connect with edges? Don't forget to connect each red vertex to its corresponding violet vertex.";
"8-cell text 2" = "Can you find the hypercube's 3D cells? How many 3D cells does it have?";
"8-cell text 3" = "Can you find the hypercube's sixteen 0D vertices, its thirty-two 1D edges and its twenty-four 2D faces?";

"AltVert title" = "What sort of 4D polyhedron\ndo alternating vertices\nof a hypercube give you?";
"AltVert text 1" = "To get warmed up, start with an ordinary 3D cube (make it all green in 4D Draw). Delete every other vertex in alternation, so only four of the cube's original eight vertices remain. Connect those four vertices with edges. What sort of polyhedron do you get?";
"AltVert text 2" = "Now try the same construction starting with a 4D hypercube. Delete every other vertex in alternation, so only eight of the hypercube's original sixteen vertices remain. Where should you put the edges, to make a convex 4D hyperpolyhedron?";
"AltVert text 3" = "Can you see this 4D hyperpolyhedron's 3D cells? How many are there? What is the shape of each one?";
"AltVert text 4" = "This 4D hyperpolyhedron's 0D vertices and 1D edges will be obvious. Can you also find its 2D faces?";
"AltVert text 5" = "Compare this 4D hyperpolyhedron to the 4D analog of an octahedron (see above). How are they similar? How are they different?";

"Diag title" = "What does a hypercube's\n“diagonal slice” look like?";
"Diag text 1" = "First consider a 2‑dimensional person's drawing of an ordinary 3D cube, if the cube sits with a long diagonal perpendicular to that person's 2D world:";
//	Image "Exercises/diagonal slice/3D cube plain" goes here
"Diag text 2" = "If the 2D world passes through the cube's center, the cross-sectional slice will be a green hexagon:";
//	Image "Exercises/diagonal slice/3D cube with slice" goes here
"Diag text 3" = "The question now becomes, if we 3‑dimensional people draw a 4D hypercube with its long diagonal perpendicular to our 3D space, and with our 3D space passing through the hypercube's center, what will the 3D cross-sectional slice be?";
"Diag text 4" = "Hint: Start by putting the “lowest” corner at (0, 0, 0, red). Add the next four corners at (‑2, ‑2, ‑2, yellow), (‑2, 2, 2, yellow), (2, ‑2, 2, yellow) and (2, 2, ‑2, yellow). Then keep adding more corners, until you've built the complete hypercube. Finally, connect the vertices at the “green” level to see what the hypercube's diagonal slice looks like. If you want, you can then delete all the hypercube's non-green vertices, so you can see the diagonal slice by itself, without distraction (but save a copy first!).";

"Checker title" = "What does a 4D fused checkerboard tiling look like?";
"Checker text 1" = "To get warmed up, start with a 2D checkerboard";
//	Image "Exercises/fused checkerboard/2D image A" goes here
"Checker text 2" = "split every other square into four triangles";
//	Image "Exercises/fused checkerboard/2D image B" goes here
"Checker text 3" = "and then fuse each triangle with the nearest whole square";
//	Image "Exercises/fused checkerboard/2D image C" goes here
"Checker text 4" = "The result is again a checkerboard, but the new squares are tilted 45° and have twice the area of the old squares.";
"Checker text 5" = "Now build yourself a 3D checkerboard (a cubic tiling, all green) and imitate the same procedure: split every other cube into six pyramids, and then fuse each pyramid with the nearest whole cube to get a new tiling of 3D space. In this new tiling, what is the shape of each 3D cell? How many 2D faces does each 3D cell have? Are those 2D faces regular or irregular (that is, are they perfect squares or merely rhombi)?";
"Checker text 6" = "Finally, build yourself a 4D checkerboard and imitate the same procedure: split every other hypercube into eight hyperpyramids, and then fuse each hyperpyramid with the nearest whole hypercube to get a new tiling of 4D space. In this new tiling, what is the shape of each 4D hypercell? How many 3D hyperfaces (“cells”) does each 4D hypercell have? Are those 3D cells regular or irregular?";
