Personal Blogs

Gaspard Monge was a French mathematician of the 18th and 19th century, a man of many parts who was a friend of Napoleon and took part in the latter's expedition to Egypt. He gave his name to a theorem in geometry which is easy to see in this illustration from Wikipedia (image by Jason Quinn).

Although it is usually presented as as theorem about circles and tangents it seems more general than that. Here is my drawing of an analogue for equilateral triangles, produced using GeoGebra.

For any pair of triangles, the lines drawn through pairs of corresponding points meet at a point, and for any set of three triangles taking the triangles in pairs produces three such points that all lie on the same line.

I don't think equilateral triangles are special, as far as I can see we could choose any shape.

Monge's theorem can also be generalised to three dimensions (four spheres and six cones, one for each pair of spheres, all of whose vertices lie on a common plane), and indeed to higher dimensions, although it obviously gets harder to visualise.

Herb and Spice-themed Daffynitions

Anise: Sibling's daughter.

Borage: Jail term.

Chervil: Type of rodent, often kept as a pet.

Chicory: Deception.

Cloves: Fings you wear.

Cumin: Opposite of going.

Fennel: Tapering tube for guiding liquids.

Mace: Small rodents.

Marjoram: Impaired your molasses-based alcoholic spirit.

Rosemary: Got up in a cheerful mood.

Sage: Reason why I'm not as sprightly as I once was.

Thyme: See Borage. 

Our old friend Tom has been at it again.

After I'd thought of that one I wondered if Chat GPT knew about Tom Swifties. It does and came up with quite a decent effort when prompted with "pie" and "Tom Swifty".

Some more geometry! 

We can inscribe a square in an equilateral triangle so that all itas corners lie on the sides of the triangle. How? Well, consider this sketch.

We draw a chord of the triangle parallel to the base and draw a square with the chord as one of its sides. Then we move the chord vertically downwards, keeping it parallel to the base and with its ends on the sides of the triangle and progressively enlarging the square, until the base of the square lands on the base of the triangle, and we are done.

We can extend the idea to three dimensions and inscribe a cube in a regular tetrahedron.

We take a section parallel to the base and in this equilateral triangle inscribe a square using the method described above, and construct a cube with this square as one of its faces. Then we move the section vertically downwards, keeping its vertices on the sides of the tetrahedron, progressively enlarging the cube, until the base of the cube lands on the base of the tetrahedron, and we are done.

It's conjectured, but I don't think proved, that this gives the largest cube that can be inscribed in a regular tetrahedron, see here.

Rather neat, and It seems to me that an analogous construction would be possible in dimension and beyond, a hypercube in a 4-simplex and so on, but I can’t prove it and drawing a picture of even the four dimensional case feels quite a challenge.

This attractive little plant likes my front garden wall, where it grows profusely. In the photo it’s intertwined with some actual ivy, at left, and you can see that the leaves of the toadflax really do look like miniature ivy leaves.

This year there is more of the toadflax than I can remember ever seeing before. Perhaps it’s because we have such a lot of rain in the last few months.

🍇

If you try to steal my grapes, then you better watch Shiraz.

Last week I visited Solva, a place in South-West Wales. It's a fishing village and harbour. Up on the cliff overlooking the sea there is an (Iron Age?) fort, it was once an significant commrercila port and centre for lime burning, and it probably has some Viking associations. The name might be derioved from Old Norse Sol = sun and Vo/Voe whichh means inlet in English and so may have had a similar meaning in Old Norse. But the orgin of the name does not seem to be attested - there is no early written evidence - so it's hard to know for sure.

Here is a picture of the estuary by Bill Boaden.

Here is a photo taken by one of our party, showing what it looks like from the shore with the tide out.

From Solva there is a cliff path that takes you to St David's and the cathedral of the monastery founded by the saint. It was a fine day and I would have liked to have walked it, but I am simply not mobile enough.

There beautiful flowers are snakes-head fritillaries, in flower a couple of weeks early than usual.

They are widely cultivated, but also grow wild in southern and central Britain, although it is not clear whether it is native or a garden escape. See Wkipedia for more on this.