May 25, 2019

Aaron Riker at Gettysburg

Aaron Denton Riker (1830-1914) of Champaign County, Ohio, enlisted as a private in the 66th Ohio Volunteer Infantry regiment on October 11, 1861. The regiment was mustered in for three years service on December 17, 1861, under the command of Colonel Charles Candy. In April 1862,  while in Strasburg, Virginia, during the Shenandoah campaign, Riker was assigned to the commissary department, handling supplies for the troops. In October, he found himself in charge of the regiment's commissary and subsequently attained the rank of sergeant while his regiment was stationed in Dumfries, Virginia.

Riker was mustered out of the regiment in 1865 as a first lieutenant.

Aaron D. Riker, Columbus, Ohio, July 27, 1865.

Riker kept a detailed journal recounting his experiences during the Civil War. The journal is now housed at the George J. Mitchell Department of Special Collections and Archives, Bowdoin College Library, Brunswick, Maine. Journal. Transcript.

Here is Riker's account of his regiment's role at the Battle of Gettysburg on Culp's Hill (edited for spelling, punctuation, and consistency). At the time, the 66th Ohio Infantry, commanded by Lieut. Col. Eugene Powell, was in the first brigade (Col. Charles Candy) of the second division (Brig. Gen. John W. Geary) of the 12th army corps (Maj. Gen. Henry W. Slocum) in the Army of the Potomac (Maj. Gen. George G. Meade).

Gettysburg, Pennsylvania, July 1863

Our army followed close on their track. Our corps crossed the river and marched to Frederick City, Maryland, thence in the direction of Gettysburg. The Rebels were evidently trying to get out of the scrape they were in and get safely back across the Potomac.

Our advance came up with them a little beyond Gettysburg, where considerable skirmishing ensued. Our advance fell back to the main body and formed in order of battle in the form of a horseshoe occupying the crest of a hill overlooking the town of Gettysburg. Our corps, the 12th, occupied the right.

The town of Gettysburg today, as seen from the observation tower atop Culp's Hill.

This was the [2nd] day of July. There had been some hard fighting today. Night coming on put an end to the battle, each army occupying good positions when the fighting ceased. Both armies slept on their arms ready to commence the work of tomorrow.

Union soldiers firing across a field in a reenactment at Spangler's Spring on the south base of Culp's Hill.

While this lull in the conflict, generals met to devise plans for the work before them. They feel the responsibility resting upon them. General Hooker has been relieved and General Meade is now in command of our army. We hope he may be successful. If not, who can foretell the consequences.

The night wears slowly away. What are the feelings of those who are watching for the coming day, knowing that with the morning's dawn, the work of death will commence? "Will I be the first victim?" may the thought of a father who thinks of a dear companion and dear little ones whom he has seen perhaps for the last time on Earth. But if he falls, may he meet them in Heaven.

Union encampment in a reenactment at Spangler's Spring on the south base of Culp's Hill.

I cannot describe this suspense before a battle. Many of our boys went into the battle of Gettysburg very low-spirited from causes which I will explain. It was thought by us and firmly believed that the Rebels were meeting with great encouragement throughout many portions of the North and that they were aided in this, their second invasion of the North, by designing and willful coworkers in treason, who prowled throughout the homes of those who were sacrificing everything, even life, to protect these homes we held most dear.

The darkest hours of the war seemed to be just preceding the battle of Gettysburg. I have heard the weather-beaten and battle-stained hero talk of despairing in the noble cause. We dreaded not the work in our front but the cowardly incendiary who invited Rebel invasion and threatened to burn and pillage in our rear.

Could we have had all this hellish band mingled with the Rebel horde--yes, every Copperhead in the North who has dared to breathe treason through his already stinking nostrils--had once been conscious they were with all their brothers in crime, the Rebel horde, and ready to meet us on the hills of Gettysburg, we could have whipped them all combined, so intense were our feelings against this cowardly class. But we know they stood a great way off and encouraged the rebellion. They would sooner make war upon helpless women and children than to face us. I fear the historian has not yet lived who can command his pen to write the infamy with which these men are stamped.

What were the feelings of our brave men, worn veterans, upon that fateful night we leave you, reader, to guess. Here was the true and tried soldier from the Keystone State; his home had been invaded, perhaps then his wife and children were being plundered and turned from their home, an aged father or mother might even then be calling for assistance, some strong arm to succor them from the hands of a relentless foe. We knew the barbarism of our enemies, and we thought too of that class who, while they were glad to take shelter under the folds of the spotless banner of Liberty, were at the same time devising means for the overthrow of our Government secretariat and openly working into the hands of Rebels in arms against us.

The morning of the [third] of July broke at last, and the heavy booming of cannon told the work of carnage had commenced. The first gun was fired by our division at 3 a.m. The fighting became general all along our lines. The constant roar of artillery, intermingled with the roar of small arms, was beyond the power of pen to describe.

Typical artillery piece on Culp's Hill.

Our forces occupied what is known as [Cemetery] Hill. The enemy made three attempts to capture this hill. They massed their columns and in solid mass attempted to carry it by storm. Our batteries discharged grape and canister into their ranks, cutting great gaps. They would waver, fall back, and repeat the charge, only to again be cut down.

The slaughter was horrible. Here our loss was also heavy. On the right [Culp's Hill], our forces were so well protected by breastworks that the loss on our part was almost nothing while that portion of the field occupied by the Rebels was literally strewn with dead.

Union soldier firing over the top of a breastwork of rocks and logs, as depicted in the monument to the 78th and 102nd New York Infantries, which took part in the defense of Culp's Hill. Such breastworks were typically surmounted by a horizontal log that protected the head but left a narrow gap to aim and fire a rifle.

But it is not my aim to give full details. Suffice it to say that night again put an end to the conflict. Thus, we celebrated the fourth of July, 1863, with our flag triumphantly waving and victory perched upon our banners. We had gained the contest. When daylight appeared in the morning of the 5th, there was no Rebel army there to give us battle. Their entire army was in full retreat, leaving us in full possession of the field.

Monument to the 66th Ohio Volunteer Infantry, positioned where the regiment fought at the top of Culp's Hill.

I visited the portion of the field where our division had fought. The loss on our side had been very light, but one man killed in the 66th and 12 wounded; the loss in other regiments but slight. Major Palmer of the 66th was mortally wounded and died a few days after the battle.

Memorial to Major J. G. Palmer marking the location where he was mortally wounded on Culp's Hill.

The loss on the side of the enemy was horrible. The dead lay in piles, in places even up to our line of breastworks. They had fearlessly charged and were mown down by dozens by our boys. Suitable parties were detailed to collect and bury the dead, and the army pursued the fleeing enemy. The 12th corps marched back by way of Frederick City on the flank of the enemy.

Principal battles in which the 66th Ohio Volunteer Infantry fought during the Civil War, based on the list engraved on the monument dedicated to the regiment in Gettysburg: Port Republic, Cedar Mountain, Antietam, DumfriesChancellorsville, Gettysburg, Lookout Mountain, Missionary Ridge, Ringgold Gap, Dug Gap, Rocky Face RidgeResaca, New Hope Church, Pine Mountain, Kennesaw Mountain, Peachtree Creek, Atlanta, Savannah.

Photos by I. Peterson, May 2019.

May 21, 2019

Art of the Tetrahedron Revisited

The tetrahedron is the simplest of all polyhedra. Any four points in space that are not all on the same plane mark the corners of four triangles. The triangles in turn are the faces of a tetrahedron.

For more than 30 years, Arthur Silverman (1923-2018) of New Orleans created artworks arising out of explorations of this angular form. "The tetrahedron is very exciting visually," Silverman explained. "It's very difficult to anticipate what you are going to see. Every step around a piece gives you a different view."

This colorful trio of tetrahedra, dubbed Painted Trio, can be found on Poydras Street in New Orleans. It's one of the few Silverman sculptures that's painted.

Located in front of City Hall in New Orleans, this sculpture resulted from removing tetrahedral forms from a rectangular block.

It's sometimes hard to tell that Silverman's sculptures are based on the tetrahedron.

Silverman created a dramatic, tetrahedron-based menorah outside Temple Sinai in New Orleans.

Changing the orientation of this Silverman sculpture, about 20 inches high, gives observers startlingly different views.

Silverman's outdoor sculptures survived the onslaught of Hurricane Katrina in 2005. Water filled his studio to a depth of about 12 inches, leaving distinct waterlines on several sculptures that had stood on the floor. Some smaller sculptures were knocked off their pedestals, resting in brackish water for days. Interestingly, several of these pieces developed a deep, rich patina--the weathered look of survivors.

Partially immersed in floodwaters for days after Hurricane Katrina in 2005, this tall Silverman sculpture displays the resulting waterline.

In 2007 at age 82, Silverman stopped creating new works. He turned his studio into a gallery where visitors could wander about and ponder the tetrahedron's amazing versatility, as unveiled by an artist's imagination.

In 2006, Silverman stopped creating new works and turned his sculpture studio into a gallery, showcasing many of his artworks.

Originally posted January 15, 2007.

Photos by I. Peterson

May 20, 2019

Arthur Silverman (1923-2018)

Arthur Silverman, who turned to sculpture after a lengthy career as a urologist in New Orleans, Louisiana, died September 24, 2018. I wrote about his use of the tetrahedron, in myriad forms, as a basis for his artworks.

In this 2007 photograph, Arthur Silverman stands with one of his tetrahedron-based sculptures. Both bent columns are identical but look quite different from different viewpoints.

Over the years, Silverman manipulated the tetrahedron's familiar shape into a wide variety of forms that were practically unrecognizable as tetrahedra.

Art of the Tetrahedron

Some might consider the tetrahedron a rather humble geometric figure.

Any four points in space that are not all in the same plane mark the corners of four triangles. The triangles in turn are the faces of a tetrahedron. It's the simplest of all polyhedra--solids bounded by polygons.

If each face is an equilateral triangle, the result is a regular tetrahedron, one of the five Platonic solids.

A regular tetrahedron, framed by a cube.

To sculptor Arthur Silverman of New Orleans, however, the tetrahedron is very special. He has been investigating variations of tetrahedral forms for more than 20 years through his sculptures, many of which are displayed in public spaces in New Orleans and other cities from Florida to California.

"The tetrahedron is very exciting visually," Silverman insists. "It's very difficult to anticipate what you are going to see."

We are accustomed to thinking about orientation in space in terms of three perpendicular axes defining left and right, up and down, and forward and backward. A regular tetrahedron has no right angles, so a tetrahedral structure can jar us out of spatial complacency. Moreover, a tetrahedron has so few faces compared to other polyhedra that its aspect changes abruptly as the observer moves to view it from different angles.  

Until age 50, Silverman had been a highly successful surgeon, practicing medicine with considerable enthusiasm and skill. Then he encountered an ailing colleague near death, who advised Silverman that if there were anything he really might want to do, then he ought to do it right away, before the chance slips away.

That encounter changed Silverman's life. He returned to interests that had captured his attention and imagination when he was a teenager. He had visited museums to gaze at statues, and had tried his own hand at carving wood. Then, when studying medicine at Tulane University, he had met a sculpture teacher who had invited him to classes and taught him how to see, in the artistic sense.

During these early explorations, Silverman had discovered the wonders of the tetrahedron, the form to which he returned with a passion many years later.

"When I first encountered tetrahedra, I was immediately fascinated by the notion of using these forms as basic building blocks for three-dimensional designs," he recalls.

So, what can you do with (or to) a tetrahedron?

You can elongate a tetrahedron, stretching several edges to a create a slim, stainless-steel tower, 60 feet high, then twin it with an identical tower in a complementary orientation to produce an elegant pair of structures, which seem to soar in formation into the sky. Such a sculpture stands in the middle of a plaza fountain in New Orleans.

Arthur Silverman's Echo features a pair of elongated tetrahedra, each balanced on one edge.

You can join tetrahedra to form a dramatic aluminum cascade.

Or you can stack them in chunks to produce an eerie column.

You can slice tetrahedra. A vast foyer wall in an office building in New Orleans is covered with aluminum tiles based on such a cross section.

The basic elements of this wall relief are sections made through a group of tetrahedra attached to each other. The changing light that plays over the wall in its foyer setting highlights different areas of the relief.

You can divide tetrahedra, then rejoin them in various ways. You can look at what's left when tetrahedra are cut out of a column or from inside a cube. You can stretch a tetrahedron and turn it inside out. You can stand it on edge or balance it on a vertex. The possibilities seem unlimited.

New Wave by Arthur Silverman. Tulane University, New Orleans, Louisiana.

"I find that the unique geometric relations intrinsic to the tetrahedron persist in the final sculpture, notwithstanding all the manipulations I carry out," Silverman notes. At the same time, "photographs do not do these works justice," he contends. "One must actually see, feel, and walk around these works in order to experience them in their totality."

The sculptor fabricates nearly all his pieces in his studio by welding together metal plates. Altogether he has produced more than 300 sculptures based on the tetrahedron, from large-scale outdoor works to diminutive studio models. "When I get an idea, I play with it as long as I can," he admits.

Arthur Silverman's sculpture of this tetrahedral form looks startlingly different from different viewpoints. Changing the sculpture's orientation gives an observer additional views.

This is art that conveys no political, social, or historical message, Silverman remarks. "The sculpture is strictly a visual experience."

One of the more intriguing of Silverman's tetrahedral creations is an ensemble of sculptures he calls Attitudes. The six pieces are spread across a grassy area at the Elysian Fields Sculpture Park in New Orleans.

All the pieces have the same geometry. Each one is made up of two identical tetrahedrons, having faces in the form of tall isosceles triangles that are welded together to form a single object. In the park, each piece has a strikingly different orientation. 

Attitudes by Arthur Silverman features the same geometric form positioned in different orientations.

When an observer walks from piece to piece, "it's hard to believe they are all the same structure," Silverman remarks. "Every time you move, you see something different."

To Nat Friedman, a mathematician and sculptor at the State University of New York in Albany, Silverman's creation is an example of a hypersculpture. Its ensemble arrangement represents a way of seeing a three-dimensional form from many different viewpoints at once.

To see every part of a two-dimensional painting in its full glory, you have to step away from it in the third dimension, Friedman says. To see a three-dimensional sculpture in its totality, you need a way to slip into the fourth dimension. Friedman calls this hypothetical process "hyperseeing." A hypersculpture consisting of a set of several related sculptures provides one way to approximate that experience.

Silverman's Attitudes, for instance, presents multiple views of an object from a single viewpoint, because copies of the same object lie in different orientations.

Originally posted Nov. 8, 1999.

Photos by I. Peterson

May 17, 2019

I. M. Pei (1917-2019)

Master architect I. M. Pei died on May 16, 2019. Among his many designs was the East Building of the National Gallery of Art in Washington, D.C.

The National Gallery of Art's East Building, designed by I.M. Pei, as seen from the rooftop of the Embassy of Canada in Washington, D.C. Opened to the public in 1978, the East Building is an eye-teasing festival of vast walls, sharp edges, odd angles, and unexpected shapes. 

Looking east toward the west-facing facade of the National Gallery of Art's East Building in Washington, D.C. Many of the building's exterior walls unexpectedly meet at acute and obtuse angles rather than commonplace right angles.

To fit the East Building on a trapezoid-shaped site, I.M. Pei based his design on a division of the trapezoid into an isosceles triangle and a smaller right triangle. Triangles serve as the structure's basic motif, as seen in the triangles of the building's ceiling and elsewhere.

Photos by I. Peterson

May 12, 2019


Together by Reza Sarhangi (1952-2016). Math-Art Exhibit, National Math Festival, Washington, D.C., 2019.

Photo by I. Peterson

May 11, 2019


Cyclides by Francesco de Corite. Math-Art Exhibit, National Math Festival, Washington, D.C., 2019.

Photo by I. Peterson

May 10, 2019

Lemon Twist

Lemon Twist by Rick Weber. Math-Art Exhibit, National Math Festival, Washington, D.C., 2019.

Photo by I. Peterson

May 9, 2019


Islamic Metamorphosis #2 by Craig Kaplan. Math-Art Exhibit, National Math Festival, Washington, D.C., 2019.

Photo by I. Peterson

May 8, 2019


Embrace by Robert Bosch. Math-Art Exhibit, National Math Festival, Washington, D.C., 2019.

See also "Moebius Meander."

Photo by I. Peterson

May 6, 2019

Temari Spherical Symmetries

Spherical Symmetries in Temari by Carolyn Yackel. Math-Art Exhibit, National Math Festival, Washington, D.C., 2019.

See also "Temari Symmetry," "Temari Ball Symmetry," and "Spherical Symmetry in Temari."

Photos by I. Peterson

May 5, 2019