• You can reflect the triangle in the minute hand, then reflect the reflection in the second hand.  How long does it take to do a complete rotation?
    You can change the triangle by dragging the red points.


  • Press Reflect twice. Now drag point I so the orange triangle lies over the image under the second reflection.  Can you describe another transformation equivalent to the two reflections?

  • Set the rotation angle and drag the center of rotation O so that the orange triangle sits on top of the image under the second reflection.  Can you describe another transformation equivalent to the two reflections?


  • A kaleidoscope is typically made with  mirrors aligned as an equilateral triangle and rotated.  Multiple reflections form a symmetrical pattern which changes as the triangle is rotated.

    Drag the corners of the four original triangles to change the pattern, then drag the red dot to rotate the kaleidoscope.

  • Here is our Kaleidoscope Click made with two triangles, the red one is the hour hand, the purple one is the minute hand.

  • In our binary digital clock, beads in the up position represent the binary digit 1, beads in the down position represent 0.  Bead positions represent binary numbers:  red beads for hours, purple for minutes and cyan for seconds.

  • Drag the beads to form the number displayed.  Drag the bead along the wire by clicking on the bead’s center point. Push the bead up  to represent the binary digit 1, drag the bead down to represent the binary digit 0.  Click the refresh button to get another number.

  • Our helpful readers have pointed out the error of our 5’s beads going the opposite way, so here is the app showing the right way to display numbers on 2 columns of an abacus.

  • The bead at the top of each column counts for 5 beads in the main compartment in that column. The beads in the main compartment of the left column each count for 10 single beads. The bead in the top compartment of the left column counts as 50.  But the top compartment is upside down in this app.  To learn the correct Abacus model, go to  Abac…[Read more]

  • It has been brought to our attention that this clock is, in some sense, upside down!  The 5’s beads in the top section are going up to indicate 5, instead of down.  To get the conventional Abacus model, go to the Abacus Corrected in this collection:  http://euclidsmuse.com/app?id=1016.

  • The form of the morph curve does depend
    on how the original curves are parametrized.
    Changing the parametrization of the original curves does not change their appearance when displayed, but it does change the appearance of the morph curve.

  • Geometrically, the morph curve is constructed by joining two corresponding points on the original curves by a line, and then positioning a third point at a specific proportion along this line.  Drag the middle red dot to see the morph curve change.

  • A morphing operation can be defined between any pair of parametric curves, open or closed. This app shows morphing a circle into a line segment.  Try moving the location of the destination line segment relative to the start of the circle’s parametrization by dragging the red dots to get different morph curves.

  • Is this clock a circle, or a square, or something in-between, or something beyond? Not only do the hands move in this clock, but its shape changes with time. At the hour it is circular, at the quarter hours it is square and at the half hour, it has gone beyond square to a concave curved square shape. The morphing is achieved by taking a linear…[Read more]

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